Effect of micromechanical models on structural responses of functionally graded plates

Accepted Manuscript Effect of Micromechanical Models on Structural Responses of Functionally Graded Plates A.H. Akbarzadeh, A. Abedini, Z.T. Chen PII: DOI: Reference:

S0263-8223(14)00477-2 http://dx.doi.org/10.1016/j.compstruct.2014.09.031 COST 5907

To appear in:

Composite Structures

Please cite this article as: Akbarzadeh, A.H., Abedini, A., Chen, Z.T., Effect of Micromechanical Models on Structural Responses of Functionally Graded Plates, Composite Structures (2014), doi: http://dx.doi.org/10.1016/ j.compstruct.2014.09.031

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Effect of Micromechanical Models on Structural Responses of Functionally Graded Plates A. H. Akbarzadeh1,2*, A. Abedini2,3, Z. T. Chen2,4 1 2 3

Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 0C3, Canada

Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

44

Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8, Canada

Abstract This paper examines the influence of various micromechanical models on the macroscopic behavior of a functionally graded plate based on classical and shear-deformation plate theories. Different micromechanical models are tested to obtain the effective material properties of a two-phase particle composite as a function of the volume fraction of particles which continuously varies through the thickness of a functionally graded plate. The static, buckling, and free- and forced-vibration analyses are conducted for a simply-supported functionally graded plate resting on a Pasternak-type elastic foundation. The volume fraction of particles are assumed to change according to the powerlaw, Sigmoid, and exponential functions. The governing partial differential equations are solved in the spatial coordinate by Navier solution, while a numerical time integration technique is employed to treat the problem in the time domain. Finally, the numerical results are provided to reveal the effect of explicit micromechanical models such as Voigt, Reuss, Hashin-Shtrikman bounds, and LRVE as well as the semi-explicit model of self-consistent on the static and dynamic displacement and stress fields, critical buckling load, and fundamental frequency. Keywords: Buckling, Elastic foundation, Functionally graded plate, Fundamental frequency, Micromechanics, Static and dynamic response.

1. Introduction

*

Address correspondence to Post-Doctoral Fellow and Honorary Research Associate Dr. A. H. Akbarzadeh: [email protected] and [email protected], Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 0C3, Canada, Tel: 514-398-6296, Fax: 514-398-7365.

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

While the anisotropic constitution of conventional laminated composites leads to delamination, matrix cracking, and adhesive bond separation due to the stress concentration and geometric discontinuities, functionally graded materials (FGMs) with a spatially continuous transition of material properties alleviate the stress concentration, optimize the structural performance, and simultaneously meet the multiphysical requirements [1, 2]. FGMs are composite materials of one or more phases dispersed in a matrix of another phases; they could be associated with particle composites where the volume fraction of particles are tailored in an arbitrary direction. Among the numerous advantages offered by FGMs, one can refer to reduced multiphysical stresses, higher fracture toughness, reduced intensity factor, and improved residual stress distribution [3, 4]. Due to the application of functionally graded (FG) thin/thick-walled structures in aerospace, pressure vessels, electronics, and medical industries, the accurate prediction of the behavior of FG structural components is of great significance [5, 6]. Shell-like structures made of composite materials and FGMs play a significant role in engineering of weight-efficient structures. To effectively describe the structural behavior, different mathematical models for shell theories have been developed [7]. Because of the computational cost of the three-dimensional (3D) elasticity analysis, lots of efforts have been devoted to develop a consistent equivalent single-layer (ESL) model for structural analysis in which the 3D structural element is replaced by an equivalent two-dimensional (2D) layer with a complex constitutive equation [1]. Since the effect of transverse shear deformation is neglected in the classical laminated plate theory (CLPT), different shear-deformation theories such as the first-order shear deformation theory (FSDT) and third-order shear deformation theory (TSDT) have been introduced. While FSDT assumes a constant shear stress through the thickness of structure and therefore needs a shear correction factor, the TSDT possesses a quadratic variation for shear stresses and thus no shear correction factor is needed [8, 9]. Since the emergence of FGMs, the structural behavior of unconstrained/constrained FG components including static, stability, and free- and forced-vibration analyses under multiphysics loading has been the subject of several theoretical and experimental investigations [10-20]. For instance, elasticity solutions were obtained in [21-23] for FG beams and plates subjected to electromechanical loading. A microstructure-dependent nonlinear beam theory, using the modified, couple stress theory, has recently been reported by Reddy [24] and the corresponding nonlinear static problem of FG beams was studied in [25] using the finite element method. Refined plate theories have also been developed in [26, 27] to accurately predict the free-vibration behavior of FG plates. A series of closed-form and semi-analytical solutions for structural responses of FG thick plates under transient thermomechanical loading and a moderately-thick variable stiffness plate were presented by Akbarzadeh et al. [28-31]. The 3D elasticity and finite element models were also given in [32, 33] for a dynamic analysis of single/multi-direction FG and sandwich plates. Moreover, to avoid the instability of structures working at different types of multiphysics loading, the buckling analysis of FG components has been conducted in several studies. For instance,

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

closed-form solutions for thermomechanical buckling of FG thin/thick plates have been presented in [34-36]. While most papers in the literature on FGMs employ the simple rule of mixture to obtain the effective material properties, a proper micromechanical model should be used to accurately predict the effective multiphysics properties. As Eshelby elucidated, the objective of micromechanics is to quantify the effect of microstructure on the multiphysics behavior of materials by the application of continuum mechanics to a small-scale [37, 38]. Among models in the literature, a few standard micromechanical models could be mentioned. Voigt’s [39] and Reuss’ [40] approximations are the simplest models used to evaluate the effective material properties of composites. Using the variational principle, Hashin and Shtrikman [41, 42] established the upper and lower bounds of the effective material properties. Mori-Tanaka [43] model was introduced to calculate the average internal stresses in the matrix containing an eigenstrain. Benveniste [44] also reformulated the Mori-Tanaka model in order to apply it to composite materials. Finally, the double inclusion methods were proposed by Lielens [45] and Nemmat-Nasser and Hori [46] based on an interpolation of the Mori-Tanaka scheme as a function of the volume fraction of the phases to predict the effective properties of composites. Several micromechanical models of FGMs have been reviewed in [47-51]. To assess the effect of the micromechanical models on the structural responses of FG plates, this paper presents the static, buckling, and free- and forced-vibration analyses for simply-supported FG plates resting on an elastic foundation. Different micromechanical models are examined to obtain the effective material properties of FGMs with power-law, Sigmoid, and exponential function distributions of volume fraction within the thickness of the plate. Using an analytical method along with a numerical time integration technique, the governing equations are treated and the effects of Voigt, Reuss, Hashin-Shtrikman bounds, LRVE, Tamura, and selfconsistent models on the structural responses of the FG plate are investigated.

2. Effective Properties of FGMs FGMs possess a continuous variation of material constituents in spatial coordinates. Such a graded microstructure could be examined as a continuous distribution of discrete particles in a matrix of a reinforced composite. The existing micromechanical models could be extended to predict the effective material properties of FGMs for an entire range of volume fraction (VF) of constituents (0 a VF a 1) [49]. Consider a two-phase FG plate composed of particles or inclusions and a matrix. While FGMs are typically made from a mixture of ceramics and metals, the material constituent could be, arbitrarily, any two dissimilar materials. The composition of two materials is assumed to vary through the thickness of the plate (z-direction, where z is downward and normal to the middle surface of the plate). The volume fraction of inclusions could vary

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

through the thickness in the form of power-law (P-FGM), Sigmoid (S-FGM), or exponential (E-FGM) [52-55]: n

¤ 2z  h ³ VF  VFt  (VFb VFt ) ¥ ´ ¦ 2h µ ª ¤ 1 ¤ h 2 z ³n ³ h 0a za ­VFt  (VFb VFt ) ¥1 ¥ ´ ´ ¥ 2¦ h µ ´ 2 ­­ ¦ µ VF  « ¤ 1 ¤ h  2 z ³n ³ ­ h ( ) VF VF VF 

aza0 ¥ ´ b t ¥ ´ ­ t ¥2¦ h µ ´ 2 ­¬ ¦ µ

(P-FGM) (1a)

(S-FGM) (1b)

¤ ¤ VF ³ ¤ 2 z  h ³n ³ VF  VFt exp ¥ ln ¥ b ´ ¥ (E-FGM) (1c) ¥ ¦ VFt µ ¦ 2h ´µ ´´ ¦ µ where VFt and VFb are, respectively, the volume fraction of inclusions at the top ( z  h / 2) and the bottom ( z  h / 2) of FG plates. Furthermore, n and h stand for the nonhomogeneity index and thickness of the plate. The non-homogeneity index n could be used to optimize the structural performance of FGMs. In this work, the Voigt, Reuss, HashinShtrikman bounds, LRVE, Tamura, and self-consistent methods are employed to obtain the effective material properties as a function of inclusion volume fraction. 2.1. Voigt and Reuss The simplest micromechanical model to achieve the equivalent macroscopic material properties is the rule of mixture which was first formulated by Voigt [39]. The Voigt idea was to determine material properties by averaging stresses over all phases with the strain uniformity assumption within the material. The Voigt model, that is frequently used in most FGM analyses, estimates Young's modulus ( E ) and Poisson's ratio (N ) of FGMs as [56, 57]:

E ( z )  EiVF ( z )  Em (1 VF ( z )) , N ( z )  N iVF ( z ) N m (1 VF ( z ))

(2) where the subscripts “i” and “m” denote the material properties of matrix and inclusions (particles). On the other hand, Reuss [39] assumed the stress uniformity through the material and obtained the effective properties as [56, 57]:

E( z) 

Ei Em N iN m , N ( z)  Ei (1 VF ( z ))  EmVF ( z ) N i (1 VF ( z )) N mVF ( z )

(3)

As shown by Hill [58], the Voigt and Reuss estimations provide, respectively, the upper and lower bounds for Young's modulus for the entire range of inclusion volume fraction. However, as Zimmerman [57] observed, Poisson's ratio could not be bounded to either the Poisson's ratios predicted by Voigt or Reuss, or even the Poisson's ratios of matrix and inclusions. It is worth mentioning that the effective mass density R is obtained by the following rule of mixture, regardless of the utilized micromechanical model:

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

R ( z )  RiVF ( z )  R m (1 VF ( z ))

(4)

2.2. Hashin-Shtrikman Bounds Using the variational principle for heterogeneous linear elasticity, Hashin and Shtrikman derived closed-form expressions for upper and lower bounds of the effective elastic properties. For two-phase materials with a random distribution of spherical particles, the bounds on effective shear (G ) and bulk ( K ) moduli are obtained as [56]: VF ( z ) VF ( z ) , K ( z)  Km  G ( z )  Gm  ª 1 ª 1 6( K m  2Gm )(1 VF ( z )) ¹ 3(1 VF ( z )) ¹   « º « º 5Gm (3K m  4Gm ) » ¬ K i K m (3K m  4Gm ) » ¬ Gi Gm

(i w m) (Lower bound) (5b) (1 VF ( z )) (1 VF ( z )) , K  ( z )  Ki  G  ( z )  Gi  ª 1 ª 1 6( K i  2Gi )VF ( z ) ¹ 3VF ( z ) ¹   « º « º 5Gi (3K i  4Gi ) » ¬ K m K i (3K i  4Gi ) » ¬ Gm Gi (i w m) (Upper bound) (5b) where superscripts “  ” and “ ” stand for the upper and lower bounds. While the lower bound is obtained when the softer phase is assumed as the matrix, the upper bound is achieved by the assumption of harder phase as the matrix. Young's modulus and Poisson's ratio are then obtained in terms of shear and bulk moduli: 9G ( z ) K ( z ) 3K ( z ) 2G ( z ) E( z)  , N ( z)  (6) G ( z )  3K ( z ) 2G ( ( z )  3K ( z )) It is worth mentioning that the effective material properties given by Hashin-Shtrikman lower bound are equivalent to the Mori-Tanaka micromechanical model. Furthermore, the upper bound can also be achieved by interchanging the matrix and inclusion in the Mori-Tanaka formulation [59]. 2.3. Tamura The method of Tamura et al. [60, 61] assumes a linear rule of mixture for effective Poisson's ratio of a two-phase composite whereas incorporates an empirical fitting parameter qT (“stress-to-strain transfer”) in the effective Young's modulus formulation. The empirical parameter relates the stress and strain in the matrix and particle phases. As a result, the effective Young's modulus and Poisson's ratio are found as: (1 VF ( z )) Em (qT Ei )  VF ( z ) Ei (qT Em ) , N ( z )  N iVF ( z )  N m (1 VF ( z )) (7) E( z)  (1 VF ( z ))(qT Ei )  VF ( z )(qT Em ) The Tamura formulation reduces to the Voigt estimation for qT  oc , and Reuss estimation for qT  0 .

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

2.4. LRVE The cubic local representative volume elements (LRVE) were also employed by Gasik and Lilius [62, 63] to obtain the effective material properties of FGMs by assuming the second phase of a two-phase composite as an inclusion. According to this micromechanical model, a locally orthotropic representative volume element could be used to predict the bulk properties of FGMs. Young's modulus is expressed as follows by the LRVE method: ¤ ³ VF ( z ) 1 E ( z )  Em ¥1  , FE  (8) ¥ FE 3 VF ( z ) ´´ 1 Em / Ei ¦ µ Furthermore, the Voigt micromechanical scheme could be used to determine the effective Poisson's ratio of FGMs. 2.5. Self-Consistent Method The self-consistent method (SCM) was originally proposed to simulate the behavior of singlephase polycrystalline materials. Nonetheless, SCM has attractive features for applications in FGMs. Since SCM does not make a distinction between particles and matrix phases, SCM is suitable for determining the effective properties of FGMs with interconnected skeletal microstructure [49, 51]. Whilst the aforementioned microstructural models, given in Sections 2.1 through 2.4, lead to an explicit formulation for effective properties, SCM provides a semiexplicit scheme. For composites with spherical inclusions in a continuous matrix, the effective properties are calculated as [48]: Km 5Gi  2 4 G z G ( )

i Km  G( z) 3 VF ( z )  (9a) Km Ki 5Gi 5Gm 

4 4 K m  G ( z ) G ( z ) Gi K i  G ( z ) G ( z ) Gm 3 3 1 4 (9b) K ( z) 

G( z) 1 VF ( z ) VF ( z ) 3  4 4 Ki  G ( z ) K m  G ( z ) 3 3 The SCM effective properties cannot be obtained directly from volume fraction and material properties of constituents; however, as mentioned by Zuiker [48], a table of effective properties for M and K could be obtained as a function of volume fraction by calculating VF as a function of M , using Eq. (9a), and then determining K , using Eq. (9b). Subsequently, Young's modulus and Poisson's ratio could be obtained by Eq. (6). The SCM model does not need empirical fitting parameters and provides a good estimation of material properties over

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

the entire range of volume fractions from zero to one. The SCM scheme is particularly beneficial when information on the microstructure of FGM is limited [48, 49, 51]. 2.6. Comparison of Micromechanical Models The estimations of Young’s modulus and Poisson's ratio of a two-phase particle reinforced composite, using the aforementioned six micromechanical models, are compared in Fig. 1. The dual-phase composite under consideration is a metal matrix composite consisting of an aluminum matrix and perfectly bonded spherical ceramic inclusions, with material properties given in Table 1. The estimated results are depicted as a function of volume fraction of inclusions (ceramic). It is worth mentioning that particle clustering (agglomeration), interfacial condition of matrix and particles, and particle size could affect the multifunctional properties of particle-reinforced composites [64]. As observed in [64-67], these parameters have significant effects on plastic deformation as well as in composites containing nanoscale particles. Considering the application of microscale particles in the elastic regime in this paper, the influence of particle clustering, interfacial parameters, and size effect on effective elastic properties are neglected. As shown in Fig. 1, Voigt and Reuss approximations plot upper and lower bounds for estimation of Young's modulus. While Reuss estimation for Young's modulus could be 48% lower than the Voigt estimation for the volume fraction of VF  0.5 , Hashin-Shtrikman bounds provides a shrinked estimation band capture by the upper and lower bounds. For instance, the estimation of Young's modulus using the lower bound (LB) Hashin-Shtrikman is 30% lower than its counterpart using the upper bound (UB) Hashin-Shtrikman. As shown in Fig. 1a, the estimates made by SCM, LRVE, and Tamura (q  100GPa ) are within the Hashin-Shtrikman bounds. It is worth noting that the estimates by Tamura greatly depend on the value of qT . Figure 1b reveals that the Poisson's ratio fails to obey the rule of mixture of Voigt or Reuss type which is consistent with Zimmerman's findings [57]. As seen in Fig. 1, a large range of effective material properties could be predicted by different micromechanical approaches. Indeed, experimental measurements are needed to select a proper micromechanical model valid in the entire range of volume fraction. In the lack of experimental data, two micromechanical requirements given by Zuiker [48] could be used to select an appropriate micromechanical model for FGMs composed of perfectly bonded spherical inclusions and a matrix. Accordingly, the predicted bulk modulus should be: (1) in the range of Hashin-Shtrikman bounds, and (2) tangent to the dilute estimation [68] at volume fractions near zero and one at which it is assumed the phase with higher volume fraction forms a continuous matrix with a dilute dispersion of the second phase with a lower volume fraction. Among the micromechanical models in this paper, SCM is the only micromechanical model that satisfies both requirements. Whilst most papers in the literature on FGM analysis employ the Voigt estimation, the numerical results in Fig. 1 show that the

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

Voigt micromechanical model overestimates Young's modulus up to 46% and Poisson's ratio up to 6% compared to the SCM. Table 1. Material properties of FGM constituents (Ceramic as an inclusion and aluminum as a matrix). Material

E (GPa)

N

R (kg / m3 )

Ceramic (Aluminum oxide 99% pure)

380 69

0.22 0.33

3.98 r103 2.71r103

Aluminum alloy 1100

(a)

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

(b) Fig. 1 Effective (a) Young's modulus and (b) Poisson's ratio as a function of volume fraction of ceramic for several micromechanical models.

3. Problem definition and governing equations To assess the effect of micromechanical schemes on the structural responses of FGM structures, we consider an FG rectangular plate with length a , width b , and thickness h resting on a Pasternak elastic foundation with the Winkler spring stiffness kw and shear layer stiffness ks , as shown in Fig. 2. The displacement field for CLPT is expressed as follows [1]: tw ( x, y, t ) u ( x , y , z , t )  u0 ( x , y , t ) z 0 tx tw ( x, y, t ) v( x, y, z , t )  v0 ( x, y, t ) z 0 ty (10) w( x, y, z , t )  w0 ( x, y, t ) The TSDT [8] has the following displacement field: tw ( x, y, t ) u ( x, y, z , t )  u0 ( x, y, t )  zJ x ( x, y, t ) c1 z 3 (J x ( x, y, t )  0 ) tx tw ( x, y, t ) v( x, y, z , t )  v0 ( x, y, t )  zJ y ( x, y, t ) c1 z 3 (J y ( x, y, t )  0 ) ty (11) w( x, y, z , t )  w0 ( x, y, t )

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

where (u , v, w) are the displacement components along ( x, y, z ) coordinate axes, (u0 , v0 , w0 ) represents the displacement components of the mid-plane ( z  0 ), and t stands for time. Furthermore, J x and J y , respectively, stand for the rotations about the y and x axes and c1 

4 . The displacement field of Eq. (11) could also be reduce to FSDT by setting c1  0 . 3h 2

Fig. 2 An FG plate resting on a Pasternak elastic foundation.

The von-Karman strains in terms of the displacement field can be written as follows for small strains and moderate rotations based on FSDT/TSDT [8]: ªE xx(3) ¹ ªE xx ¹ ªE xx(0) ¹ ªE xx(1) ¹ (0) (2) ­ ­ ­ (0) ­ ­ (1) ­ 3 ­ (3) ­ ªG yz ¹ ªG yz ¹ 2 ªG yz ¹   z , (12) z z E E E E    « (2) º « yy º « yy º « yy º « yy º « º « (0) º ¬G xz » ­G ­ ­G (0) ­ ­G (1) ­ ­G (3) ­ ¬G xz » ¬G xz » ¬ xy » ¬ xy » ¬ xy » ¬ xy » where (E xx , E yy ) and (G xy , G xz , G yz )  2(E xy , E xz , E yz ) are the normal and transverse strains, respectively, and: 1 ª ¹ u0, x  w0, x 2 ­ ­ (1) (3) 2 ªE xx(0) ¹ ­ ª J x , x  w0, xx ¹ ­ ªE xx ¹ ª J x , x ¹ ªE xx ¹ 1 ­ (0) ­ ­ ­ ­ (3) ­ ­ ­ ­ ­ (1) ­ ­ 2 «E yy º  « v0, y  w0, y º , «E yy º  « J y , y º , «E yy º  c1 « J y , y  w0, yy º 2 ­G (0) ­ ­ ­ ­ (1) ­ ­ ­ ­ (3) ­ ­J  J  2 w ­ y,x 0, xy » ¬ xy » ­u0, y  v0, x  w0, x w0, y ­ ¬G xy » ¬J x , y  J y , x » ¬G xy » ¬ x, y ­ ­ ¬ » (0) ªG yz ¹ ªJ y  w0, y ¹ ªG yz(2) ¹ ªJ y  w0, y ¹ (13) « (0) º  « º , « (2) º  c2 « º ¬J x  w0, x » ¬G xz » ¬J x  w0, x » ¬G xz » and c2  3c1 ; a comma denotes the partial differentiation operator. The strain-displacement equations (12) and (13) can be further reduced to those for CLPT by setting c1  c2  0 and substituting J x  w0, x and J y  w0, y .

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Using the principle of virtual displacement, equations of motion for plates resting on the Pasternak elastic foundation could be obtained. The detailed derivation procedure could be found in [1]. The equations of motion for CLPT are: 0, x N xx , x  N xy , y  I 0uo I1w

0, y N xy , x  N yy , y  I 0 vo I1w M xx , xx  2 M xy , xy  M yy , yy  Nˆ ( w0 )  q ( x, y ) k w w0  k s ( w0, xx  w0, yy )  o I 2 ( w 0, xx  w 0, yy )  I1 (u0, x  v0, y ) I0 w

(14)

while the equations of motion are expressed as follows for FSDT/TSDT [15, 29]: 0, x N xx , x  N xy , y  I 0uo  J1Jx c1 I 3 w

0, y N xy , x  N yy , y  I 0 vo  J1Jy c1 I 3 w Qx , x  Qy , y  c1 ( Pxx , xx  2 Pxy , xy  Pyy , yy )  Nˆ ( w0 )  q ( x, y ) k w w0  k s ( w0, xx  w0, yy )  o c12 I 6 ( w 0, xx  w 0, yy )  c1 ( I 3 (u0, x  v0, y )  J 4 (Jx , x  Jy , y )) I0 w

0, x M xx , x  M xy , y Qx  J1uo  K 2Jx c1 J 4 w 0, y M xy , x  M yy , y Qy  J1vo  K 2Jy c1 J 4 w

(15)

where h 2

( NAB , M AB , PAB ) 

h 2 3

¯ S AB (1, z, z )dz , (QA , RA )  c ¯ S A f

h 2

z

(1, z 2 )dz , M AB  M AB c1 PAB ,

h 2

h 2

QA  QA c2 RA , I i 

i

¯ R z dz ,

J i  I i c1 I i  2 , K 2  I 2 2c1 I 4  c12 I 6 (i  0,1,..., 6)

h 2

ªc2 w 0, c f  1 TSDT « FSDT ¬c2  0

(16)

where S AB is the stress components ( A , B = x , y ). Furthermore, c f , R , and q , respectively, stand for the shear correction factor, mass density, and transverse load on the plate. The superposed dot on variables also denotes the time derivation. It is worth mentioning that the following nonlinear in-plane force resultant Nˆ ( w0 ) is used for bifurcation buckling analysis, while this term is omitted for the static as well as free- and forced-vibration analyses [1]: (17) Nˆ ( w0 )  ( N xx w0, x  N xy w0, y ), x  ( N xy w0, x  N yy w0, y ), y The stress resultants N, M, P, Q, and R, defined in Eq. (16), are also related to strains as:

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

ª [ N ] ¹ §; A= ­ ­ ¨ «[M ]º  ¨; B = ­ [ P] ­ ¨; E = ¬ » ©

; B = ; E = ¶ ­[E ]­ ; D = ; F = ·· «­[E (1) ] º­ ; F = ; H =¸· ­­[E (3) ]­­ ¬ » ª

(0)

¹

; D =¶ ª­[G ]¹­ ; F =·¸ ­¬«[G (2) ]­»º

(18a)

(0)

ª­[Q]¹­ § ; A= « º¨ ­¬[ R]­» ©; D =

(18b)

where the stiffness matrices are defined as: h 2

( Aij , Bij , Dij , Eij , Fij , H ij ) 

¯ Q (1, z, z

2

ij

, z 3 , z 4 , z 6 )d

(19)

h

2

in which Qij is the symmetric, transformed, and plane stress-reduced stiffness. The non-zero components of Qij for FG plates are: Q11 ( z )  Q22 ( z ) 

N ( z)E( z) E( z) , Q12 ( z )  , and 2 1 N ( z ) 1 N ( z ) 2

E( z) . By substituting Eqs. (12), (13), (17), and (18) into Eqs. 2(1 N ( z )) (14) and (15), the governing differential equations of motion for CLPT and TSDT are derived, respectively. The nonlinear strain-displacement terms and nonlinear in-plane force resultant, Nˆ ( w0 ) , are omitted in governing differential equations for the linear static and freeand forced-vibration analyses, while Nˆ ( w ) is preserved for the bifurcation buckling analysis. Q44 ( z )  Q55 (z)  Q66 ( z ) 

0

4. Methodology To provide an analytical methodology for structural responses of FGM plates in spatial coordinate, the moveable simply-supported boundary conditions (SS-1) are considered in this paper. The following displacement and stress resultant boundary conditions are assumed for FSDT/TSDT [1]: u0 ( x, 0, t )  0 , u0 ( x, b, t )  0 , v0 (0, y, t )  0 , v0 (a, y, t )  0

J x ( x, 0, t )  0 , J x ( x, b, t )  0 , J y (0, y, t )  0 , J y (a, y, t )  0 w0 (0, y, t )  0 , w0 (a, y, t )  0 , w0 ( x, 0, t )  0 , w0 ( x, b, t )  0 N xx (0, y, t )  0 , N xx (a, y, t )  0 , N yy ( x, 0, t )  0 , N yy ( x, b, t )  0 (20) M xx (0, y, t )  0 , M xx (a, y, t )  0 , M yy ( x, 0, t )  0 , M yy ( x, b, t )  0 For the forced-vibration analysis, zero initial conditions are adopted for displacements. Neglecting the rotation terms J x and J y in Eq. (20) along with setting c1  c2  0 provide the boundary conditions for CLPT analysis. The Navier [1, 30] solutions could be developed for the FGM plate with SS-1 boundary conditions (Eq. 20). Consequently, the displacement

12

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

fields are given in the following trigonometric form using the Fourier series expansion for FSDT/TSDT [30]: ª u0 ( x , y , t ) ¹ ªU mn (t ) cos(rm x) sin(rn y ) ¹ ­ v ( x, y , t ) ­ ­ ­ ­­ 0 ­­ c c ­­ Vmn (t ) sin(rm x) cos(rn y ) ­­ (21) « w0 ( x, y, t ) º  £ £ « Wmn (t ) sin(rm x) sin(rn y ) º ­J ( x, y, t ) ­ n 1 m 1 ­ X (t ) cos(r x) sin(r y ) ­ m n ­ x ­ ­ mn ­ ­¬J y ( x, y, t ) »­ ¬­ Ymn (t ) sin(rm x) cos(rn y ) »­ mP nP and rn  ; U mn , Vmn , Wmn , X mn , and Ymn are unknown coefficients to be a b determined. Since the FGM plate is subjected to the transverse load q for the static and forced-vibration analyses, q should also be expanded as:

where rm 

c

c

q ( x, y, t )  ££ Qmn (t) sin(rm x) sin(rn y )

(22)

n 1 m 1

For some typical transverse loads, Qmn is given in Table 2. Table 2. Coefficients in the double trigonometric series expansion of different loading types. Coefficient Qmn

Loading type Uniform

16q0 P 2 mn

q ( x, y, t )  q0 (t ) Partially uniform

q ( x, y, t )  q0 (t ) for a1 a x a a2 and b1 a y a b2

4q0 (cos(rm a2 ) cos(rm a1 ))(cos(rnb2 ) cos(rnb1 )) P 2 mn

Point load1

4Q0 sin(rm x0 ) sin(rn y0 ) Lx Ly

Q( x, y, t )  Q0 (t ) at ( x0 , y0 ) 1

A moving point load is defined by: Q ( x, y, t )  Q0 (t )D ( x xmv (t ), y ymv (t )) , where xmv (t ) and ymv (t ) are the location of moving load.

Using the Fourier series expansion defined in Eqs. (21) and (22), coupled time-dependent equations of motion (14) and (15) can be rewritten in the following systems of differential equations for CLPT: CLPT CLPT CLPT  CLPT §© K mn ¶¸ ¶¸ (23) $ CLPT  §© M mn $  Fmn mn mn 3r3

[

]

[

]

3r3

3r1

and for FSDT/TSDT: TSDT §© K mn ¶¸ $TSDT mn 5r5

where

[$

CLPT mn

T

]  [U

mn

Vmn Wmn ]

5r1

TSDT ¶¸  §© M mn

and

5r5 T TSDT mn

[

]

3r1

[$ ] TSDT mn

[$ ]  [U

mn

5r1

[

]

[

]

TSDT  Fmn

Vmn Wmn

3r1

5r1

X mn

(24) Ymn ] . The

components of stiffness matrix, K , mass matrix M , and force vector F are given in Appendices A and B for CLPT and FSDT/TSDT, respectively. The system of differential

13

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

equations (23) and (24) are considered for static, free-vibration, and forced-vibration analyses in the absence of nonlinear in-plane force Nˆ ( w0 ) . The eigenvalue solution and the Newmark integration technique are adopted for free- and forced-vibration analysis, respectively [1]. In the presence of Nˆ ( w0 ) , the critical buckling load is obtained by bifurcation analysis [1]. Since the Navier solution is considered for structural analysis, only uniaxial and biaxial buckling loads could be achieved. Hence, the plate could be subjected to in-plane forces N xx0 and/or N yy0 ( N xy0  0) . It is also worth mentioning that the structural damping in forcedvibration analysis could also be taken into account by adopting the Rayleigh damping as: CLPT CLPT CLPT TSDT TSDT TSDT [Cmn ]3r3  a0 [ M mn ]3r3  a1[ K mn ]3r3 and [Cmn ]5r5  a0 [ M mn ]5r5  a1[ K mn ]5r5 (25) where a0 and a1 are Rayleigh constants which can be obtained experimentally .

5. Results and Discussion In this section, the structural responses of FG square plates are estimated by several micromechanical models. The FG plate is composed of a ceramic and aluminum alloy (Table 1) with a continuous transition of the volume fraction (VF) of ceramic through the thickness. The top of the plate is aluminum alloy rich while the bottom side is ceramic rich. The effective material properties, given in Section 3, are used in the developed methodology in Section 5. The impact of the micromechanical models, elastic foundation, non-homogeneity index, and FGM profile on the static and dynamic deflection, critical buckling load, and fundamental frequency are examined, respectively. The x- and y- coordinates, time, moving point load velocity (V) , deflection, critical buckling load ( N cr ) , fundamental frequency, von Mises stress (S VM ) , and weight are given in the following non-dimensional form: x

t x y , y , t  a a b

Em

Rm

Em h3 a2 a2 N  N  W W , , cr h Rm Em h3 a 4 q0 S weight S VM  VM , wt  R m abh q0

, V V

Em

, w  w0

Rm Em (26)

5.1. Effect of Micromechanical Models The impact of the micromechanical models on the estimated structural responses of unconstrained FG plates, with a P-FGM material profile, are studied in this section. The maximum non-dimensional out-of-plane deflection occurring at the plate midpoint, critical uniaxial buckling load, and fundamental frequency are given, respectively, in Tables 3.1 through 3.3 for a set of length-to-thickness ratios ( a h ) and non-homogeneity indices. A uniformly distributed transverse load has been considered for the bending analysis. The numerical results are obtained by using TSDT and CLPT along with Voigt, Reuss, Hashin

14

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

UB, Hashin LB, LRVE, Tamura (qT  100GPa ) , and SCM micromechanical models. The results obtained in Table 3 for the Voigt micromechanical model are reduced to those reported in [28, 52] for static bending, [69, 70] for free-vibration, and [1, 71] for buckling analyses. As shown in Table 3, the discrepancy between the results predicted by TSDT and CLPT dramatically diminishes by increasing the length-to-thickness ratio. For a moderately thick (a h  5) , homogeneous ceramic (n  c) plate, discrepancies up to 16%, 24%, and 10% are respectively observed between the estimated results by TSDT and CLPT for maximum deflection, critical uniaxial buckling load, and fundamental frequency. On the other hand, increasing the length-to-thickness ratio to a h  200 decreases the discrepancies by 2% for maximum deflection and less than 0.1% for critical buckling and fundamental frequency. While the length-to-thickness ratio plays a major role on the accuracy of CLPT results compared to the TSDT, the discrepancy between the TSDT and CLPT structural responses also depend on the material composition. For instance, comparing the TSDT and CLPT results, increasing the non-homogeneity index in an FG plate from n  1 to n  10 increases the discrepancy from 12% up to 21% for the estimated maximum deflection. The non-homogeneity index (n) of the volume fraction distribution alters the material composition of FGMs and, as a result, affects the structural behavior of FG structures. The volume fraction of ceramic in the dual-phase FGM decreases by increasing n . Due to the higher Young's modulus and lower Poisson's ratio of ceramic compared to aluminum alloy (Table 1), the maximum deflection of the FG plate is increased and the critical buckling load and fundamental frequency are reduced by increasing the non-homogeneity index. As seen in Table 3, increasing n from 0.5 to 2 in a thin FG plate (a h  200) drastically increases the maximum deflection by 64% and decreases the critical uniaxial load and fundamental frequency by 39% and 17%, respectively, predicted by the Voigt micromechanical scheme.

15

a 5 h

a  100 h

a  10 h

a 5 h

8.6209

nc

7.6581 5.9343 3.6921

6.3768 4.4905 2.9895

n  0.5 n 1 9.7521

12.5797

10.6123

8.2480

19.0406

15.8412

n2 n  10

25.8149 33.16 43.4475

22.6221

18.8949

8.4241

27.3563 35.3068 45.7670

23.7772

19.7509

8.8105

32.0171 41.7976 52.7839

27.2708

23.1595

26.4230 33.9279 44.4589

23.1595

19.3436

8.6209

26.4230 33.9279 44.4589

23.1595

19.3436

8.6209

26.4230 33.9279 44.4589

24.0385 30.8511 43.4475

20.4940

16.5342

8.4241

25.3981 32.9562 45.7670

21.4750

17.2617

8.8105

29.5094 39.3203 52.7839

24.4424

19.4628

9.9796

24.6096 31.5664 44.4589

20.9825

16.9256

8.6209

24.6096 31.5664 44.4589

20.9825

16.9256

8.6209

24.6096 31.5664 44.4589

20.9825

16.9256

8.6209

Hashin (LB) TSDT CLPT

22.3531 28.6518 43.4475

18.2838

14.3359

8.4241

23.4578 30.6876 45.7670

19.0641

14.9324

8.8105

26.7988 36.8419 52.7839

21.4247

16.7373

9.9796

22.8888 29.3179 44.4589

18.7196

14.6739

8.6209

22.8888 29.3179 44.4589

18.7196

14.6739

8.6209

22.8888 29.3179 44.4589

18.7196

14.6739

8.6209

Hashin (UB) TSDT CLPT

23.8481 30.5188 43.4475

20.3847

16.2616

8.4241

25.1790 32.6231 45.7670

21.3197

16.9479

8.8105

29.2035 38.9849 52.7839

24.1483

19.0247

9.9796

24.4164 31.2263 44.4589

20.8721

16.6468

8.6209

24.4164 31.2263 44.4589

20.8721

16.6468

8.6209

24.4164 31.2264 44.4589

20.8721

16.6468

8.6209

LRVE TSDT CLPT

23.7842 30.5116 43.4475

20.1690

16.1918

8.4241

25.096 32.6065 45.7670

21.1072

16.8889

8.8105

29.063 38.9400 52.7839

23.9452

18.9980

9.9796

4.9214 3.7655 2.9895

5.7897

7.0776

15.8412

6.2143 4.8381 3.6921

7.0908

8.4890

19.0406

Reuss TSDT CLPT

5.3436 3.9997 2.9895

6.4648

8.1266

15.8412

6.6734 5.2004 3.6921

7.8269

9.7010

19.0406

Hashin (LB) TSDT CLPT

5.8941 4.2662 2.9895

7.3851

9.4558

15.8412

7.1760 5.5997 3.6921

8.7725

11.1882

19.0406

Hashin (UB) TSDT CLPT

5.4005 4.0335 2.9895

6.54682

8.31763

15.8412

6.7266 5.2570 3.6921

7.8687

9.8634

19.0406

LRVE TSDT CLPT

5.4274 4.0384 2.9895

6.6012

8.3275

15.8412

6.7446 5.2582 3.6921

7.9529

9.9060

19.0406

Tamura TSDT CLPT

24.3500 31.2192 44.4589

20.6500

16.5749

8.6209

24.3500 31.2192 44.4589

20.6500

16.5749

8.6209

24.3500 31.2192 44.4589

20.6500

16.5749

8.6209

Tamura TSDT CLPT

Table 3.2. Non-dimensional critical uniaxial buckling load N estimated by several micromechanical models.

21.4493 27.6679 44.4589

16.8376

Voigt TSDT CLPT

20.9452 27.0369 43.4475

n0

nc

n2 n  10

16.4470

8.6209 13.0495

8.4241

12.7502

n0

nc

n  0.5 n 1

16.8376 21.4493 27.6679 44.4589

17.1282

13.0495

8.8105

13.2870

21.4493 27.6679 44.4589

16.8376

24.7938 34.9869 52.7839

19.1894

21.9014 29.0130 45.7670

n2 n  10

n  0.5 n 1

n2 n  10 nc n0

19.3436

8.6209

9.9796

13.0495

22.3403

9.9796

14.9113

n0

n  0.5 n 1

8.6209

Reuss TSDT CLPT

Voigt TSDT CLPT

Table 3.1. Maximum non-dimensional deflection ( wr1000) under uniform static load estimated by several micromechanical models.

8.6209

23.7060 30.3062 44.4589

20.0686

15.7956

8.6209

23.7060 30.3062 44.4589

20.0686

15.7956

8.6209

23.7060 30.3062 44.4589

20.0686

15.7956

5.5966 4.1357 2.9895

6.8598

8.8130

15.8412

6.9288 5.4168 3.6921

8.1839

10.3946

19.0406

SCM TSDT CLPT

23.1519 29.6190 43.4475

19.5991

15.4303

8.4241

24.4043 31.7046 45.7670

20.4585

16.0592

8.8105

28.1918 38.0099 52.7839

23.0585

17.9621

9.9796

SCM TSDT CLPT

a  100 h

a  10 h

a 5 h

a  100 h

a  10 h

n  0.5 n 1

n  0.5 n 1

CLPT

n  0.5 n 1

9.7234

8.8277

8.0824 7.4913 6.0342

n2 n  10 nc 8.0844 7.4943 6.0360

8.8297

9.7256

11.3109

11.3080

n0

n  0.5 n 1

nc

8.0089 7.4243 5.9875

7.8215 7.1517 5.8237

n2 n  10

8.7542

9.6468

9.4370

8.5671

11.2199

10.9548

n0

nc

7.7910 7.2220 5.8472

7.1877 6.3804 5.3172

8.5358

9.4189

10.9571

n2 n  10

7.9227

8.7277

n  0.5 n 1

TSDT

10.0885

6.2103 4.8346 3.6899

7.0868

8.4848

19.0310

5.8304 4.5156 3.4868

6.7128

8.0851

18.1238

6.2143 4.8381 3.6921

7.0908

8.4890

19.0406

6.2143 4.8381 3.6921

7.0908

8.4890

19.0406

6.6692 5.1964 3.6899

7.8227

9.6962

19.0310

6.2815 4.8364 3.4868

7.4344

9.2520

18.1238

6.6734 5.2004 3.6921

7.8269

9.7010

19.0406

6.6734 5.2004 3.6921

7.8269

9.7010

19.0406

7.2802 6.7644 6.0342

7.5270

7.9873

11.3080

6.9987 6.4889 5.8237

7.2673

7.7349

10.9548

6.3369 5.8474 5.3172

6.6432

7.1185

10.0885

TSDT

7.2826 6.7669 6.0360

7.5291

7.9893

11.3109

7.2176 6.7091 5.9875

7.4621

7.9193

11.2199

7.0298 6.5423 5.8472

7.2682

7.7166

10.9571

CLPT

Reuss

7.5444 7.0130 6.0342

7.9082

8.5385

11.3080

7.2623 6.7147 5.8237

7.6475

8.2757

10.9548

6.5952 6.0267 5.3172

7.0157

7.6305

10.0885

TSDT

7.5468 7.0156 6.0360

7.9103

8.5406

11.3109

7.4775 6.9544 5.9875

7.8399

8.4676

11.2199

7.2771 6.7778 5.8472

7.6362

8.2564

10.9571

CLPT

Hashin (LB)

7.1760 5.5997 3.6921

8.7725

11.1882

19.0406

7.17600 5.59969 3.6921

8.77254

11.1882

19.0406

6.7224 5.2530 3.6899

7.8647

9.8588

19.0310

6.3366 4.8855 3.4868

7.4898

9.4246

18.1238

6.7266 5.2570 3.6921

7.8687

9.8634

19.0406

6.7266 5.2570 3.6921

7.8687

9.8634

19.0406

6.7405 5.2543 3.6899

7.9488

9.9013

19.0310

6.3579 4.8880 3.4868

7.5648

9.457

18.1238

7.8237 7.2771 6.0342

8.3726

9.1698

11.3080

7.5569 6.9562 5.8237

8.1188

8.9005

10.9548

6.9152 6.2222 5.3172

7.4944

8.2334

10.0885

TSDT

7.8258 7.2790 6.0360

8.3745

9.1719

11.3109

7.7527 7.2143 5.9875

8.3012

9.0959

11.2199

7.5417 7.0248 5.8472

8.0892

8.8759

10.9571

CLPT

Hashin (UB)

7.5745 7.0510 6.0342

7.9293

8.6098

11.3080

7.2932 6.7486 5.8237

7.6751

8.3524

10.9548

6.6273 6.0524 5.3172

7.0561

7.7172

10.0885

TSDT

7.5768 7.0537 6.0360

7.9313

8.6118

11.3109

7.5063 6.9920 5.9875

7.8601

8.5382

11.2199

7.3025 6.81379 5.8472

7.6542

8.3253

10.9571

CLPT

LRVE

6.7446 5.2582 3.6921

7.9529

9.9060

19.0406

6.7446 5.2582 3.6921

7.9529

9.906

19.0406

7.5847 7.0519 6.0342

7.9716

8.6283

11.3080

7.3059 6.7503 5.8237

7.7141

8.3671

10.9548

6.6446 6.0556 5.3172

7.0879

7.7238

10.0885

TSDT

7.5870 7.0545 6.0360

7.9737

8.6303

11.3109

7.517 6.9927 5.9875

7.9028

8.5569

11.2199

7.3149 6.8144 5.8472

7.6978

8.3444

10.9571

CLPT

Tamura

estimated by several micromechanical models.

7.1721 5.5953 3.6899

8.76841

11.1831

19.0310

6.80511 5.19273 3.4868

8.37822

10.6973

18.1238

W

17

Table 3.3. Non-dimensional fundamental frequency

7.6581 5.9343 3.6921

Voigt

7.6542 5.9295 3.6899

n0

nc

n2 n  10

9.7521

12.5797

9.7477

19.0406

19.0310

12.5739

n0

nc

7.6581 5.9343 3.6921

7.2910 5.4916 3.4868

n2 n  10

9.7521

12.5797

9.3261

19.0406

18.1238

12.0216

n0

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

7.6875 7.1573 6.0342

8.0867

8.8386

11.3080

7.4075 6.8448 5.8237

7.8353

8.5813

10.9548

6.7422 6.1282 5.3172

7.2199

7.9434

10.0885

7.6898 7.1601 6.0360

8.0887

8.8407

11.3109

7.6173 7.0967 5.9875

8.0160

8.7658

11.2199

7.4077 6.9136 5.8472

7.8058

8.5493

10.9571

CLPT

6.9288 5.4168 3.6921

8.1839

10.3946

19.0406

6.9288 5.4168 3.6921

8.1839

10.3946

19.0406

SCM TSDT

6.9246 5.4126 3.6899

8.1799

10.3899

19.0310

6.5386 5.0264 3.4868

7.8064

9.9475

18.1238

While most of the research in the literature employs the Voigt micromechanical model for multiscale simulation of structural response of FG components, Table 3 reveals the significance of a proper selection of the micromechanical scheme to accurately predict the responses of FG structures. In accordance with the effective material properties given in Fig. 1 for several micromechanical models, it is observed that the Voigt and Reuss models provide the lower and upper bounds of the maximum deflection and conversely the upper and lower bounds of the critical uniaxial buckling load and fundamental frequency. Nevertheless, more accurate bounds of response domain of FG structures are drawn by using the Hashin LB and Hashin UB micromechanical models. The SCM, as the only micromechanical model which satisfies the both Zuiker [48] micromechanical requirements, predicts the structural responses within the Hashin bounds. For a moderately thick FG plate (a / h  5) with the nonhomogeneity index n  0.5 , the Hashin LB and Hashin UB predicts the maximum deflection 31% and 12% higher than the Voigt model, respectively. In contrast, buckling load predicted by the Hashin LB and Hashin UB is, respectively, 23% and 11% lower than the Voigt estimation; the discrepancy between the Hashin LB/UB and Voigt estimation for fundamental frequency is further reduced to 13% and 6%. Moreover, it is found that the SCM, Tamura, and LRVE estimate the structural responses within the domain responses of Hashin bounds, For example, the maximum deflection estimated by the SCM, Tamura, and LRVE are 20%, 27%, and 28% higher than the Voigt estimation. It is also worthwhile to note that the estimated structural responses by Tamura is remarkably dependant on the empirical fitting parameter, qT . The variation of qT from 0 to oc could provide a structural response domain bounded by the Reuss and Voigt estimations. The discrepancy between the estimated structural responses of FGMs by the Voigt and other micromechanical models depends considerably on the non-homogeneity index n . As a result, Fig. 3 is plotted to show the percentage of relative difference observed in maximum deflection, critical buckling load, and fundamental frequency of an FG plate with a / h  10 estimated by the Hashin bounds and Voigt micromechanical model for nonhomogeneity index 0 a n a 10 . While no discrepancy is observed among the estimated structural responses for the homogeneous plate (n  0 and n  c) , the Hashin LB and Hashin UB could, respectively, estimate the maximum deflection up to 30% and 12% higher than the Voigt model and the buckling load up to 23% and 11% lower than the Voigt model for n  0.5 . Therefore, the necessity of the proper micromechanical modeling of FGMs is evident to accurately estimate the structural response. Due to the significance of appropriate micromechanical modeling for structural analysis of FG structures, the SCM is employed in the following sections for investigation of the effect of FGM profile and elastic foundation on the structural response.

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

Fig. 3 Percentage of the relative difference between estimated structural responses of an FG plate by Hashin UB/LB and Voigt micromechanical models.

5.2. Effect of FGM Profile The FGM profile of dual-phase FG plates determines the distribution function of volume fraction of inclusions in a matrix, as seen in Eq. (1). To examine the effect of FGM profile on the structural performance of FG structures, the non-dimensional maximum deflection, maximum von Mises stress, critical uniaxial buckling load, fundamental frequency, and weight of a square FG plate with a / h  10 are compared in Fig. 4 for the P-FGM, S-FGM, and E-FGM profiles. The numerical results, based on the TSDT, are presented for nonhomogeneity index 0 a n a 10 . As seen in Fig. 4, increasing the non-homogeneity index, n , in P-FGM and E-FGM profiles remarkably enhances the maximum deflection and von Mises stress of the FG plate while decreases the buckling load, fundamental frequency, and weight. Hence, there is a trade-off between the structural responses and the weight of FG plates, which stimulates the application of a multi-objective optimization scheme in order to reach the optimum structural performance of FG plates. The variation of volume fraction by the non-homogeneity index in the E-FGM is more considerable than the P-FGM; for instance, while increasing the non-homogeneity index from 0 to 10 decreases the weight of a P-FGM plate by 29%, the decrease in the weight is 31% for an E-FGM plate. Consequently, the maximum deflection and von Mises stress in the E-FGM plate with n  10 is respectively, up to 22% and 4% higher than the P-FGM plate, whereas the buckling load and fundamental frequency are lower by 18% and 9%.

19

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

On the other hand, the anti-symmetric nature of S-FGM profile given in Eq. (1b) leads to the weight independence of the S-FGM plate from the non-homogeneity index in Fig. 4. Therefore, the variation of structural response of S-FGM plates for non-homogeneity index 0 a n a 10 is not remarkable compared to the P-FGM and E-FGM profiles. The variation of n from 0 to 10 could not alter maximum deflection, buckling load, and fundamental frequency higher than 11%, 10%, and 5%, respectively. It is worth mentioning that, in contrast to the P-FGM and E-FGM, n  0 leads to the constant ceramic volume fraction 0.5 within the FG plate and n  c separates the plate into two homogeneous parts of ceramic and aluminum alloy. From the aforementioned observations, the significance of selecting a proper FGM profile to satisfy the deformation, buckling, and frequency requirements of a FG plate is evident.

(a)

(b)

(c)

(d)

(e)

20

A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

Fig. 4 Effect of FGM profile on non-dimensional (a) maximum deflection, (b) maximum von Mises stress, (c) critical uniaxial buckling load, (d) fundamental frequency, and (e) weight of FG plates for a range of nonhomogeneity index 0 a n a 10 .

5.3. Effect of Elastic Foundation The effect of the elastic foundation on the estimated static, buckling, and free- and forcedvibration behavior of a P-FGM square plate is reported in this section with a length-tothickness ratio a / h  10 . The SCM micromechanical model and TSDT are used to accurately predict the static and transient responses. Four different foundation conditions are considered in this examination: (k w , k s )  (0, 0) (unconstrained), (kw , k s )  (500, 0) (Winkler),

(kw , ks )  (0,50) (shear layer), and (kw , ks )  (500,50) (Pasternak) [72]. Figure 5 illustrates the effect of foundation condition on the static bending of the FG plate with n  2 subjected to a point load (Q0  abq0 ) at the center of the plate. The numerical results based on TSDT are estimated by the Voigt and SCM. The out-of-plane deflection at the middle of plate ( y  b / 2) , along the length of the plate, is shown in Fig. 5a while the static bending stress distribution through the thickness of the plate at the plate midpoint is plotted in Fig. 5b. As seen in Fig. 5, the presence of elastic foundation decreases the out-ofplane deflection, compressive bending stress at the top of the plate, and the tensile bending stress at the bottom of the plate. Considering the shear layer effect in the Pasternak-type elastic foundation, compared to the Winkler-type elastic foundation, it further decreases the deflection and absolute value of bending stress through the plate thickness. Moreover, the numerical results reveal that the Voigt micromechanical model, independent of elastic foundation condition, underestimates the deflection and overestimates the bending stress of the FG plates compared to the SCM. However, the elastic foundation condition could alter the relative percentage difference between the Voigt and SCM estimations. The Pasternak-type elastic foundation compared to an unconstraint condition, for example, decreases the percentage difference from 11% to 6%. As a result, selection of micromechanical models could considerably affect the estimated static responses of FGMs depending on the foundation condition.

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(a)

(b) Fig. 5 (a) Non-dimensional deflection and (b) bending stress of an FG plate subjected to a point load for several elastic foundation conditions.

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Figure 6 shows the influence of elastic foundation on the dynamic response of FG plate estimated by the SCM and TSDT. The square FG plate, considered in Fig. 5, is subjected to two different dynamic transverse loads including: (a) a step-function dynamic uniform load at the midpoint of the plate, and (b) a moving point load traveling at the middle of plate ( y  b / 2) along the length of the plate with non-dimensional velocity V  0.04 . Figure 6a shows the temporal evolution of the out-of-plane deflection at the plate midpoint. Due to the neglected structural damping in the dynamic analysis, the oscillation of the midpoint dynamic deflection around the static deflection is not dissipated. As seen in Fig. 6a, the presence of elastic foundation dramatically decreases the amplitude and increases the frequency of dynamic oscillation. The application of a Pasternak-type elastic foundation, (kw , ks )  (500,50) , decreases the amplitude of forced-vibration up to 70% and increases the vibration frequency up to 81%. The elastic foundation has similar influence on the dynamic behavior of plates subjected to a moving transverse load. Figure 6b shows that the temporal evolution of deflection at the plate midpoint possesses two response domains including forced ( t a texit  25) and free ( texit a t ) vibration parts. A moving transverse point load is applied at the top of the plate and causes the forced-vibration up to the exit time of the point load texit . Once the moving point load leaves the plate, the FG plate experiences the free-vibration caused by the inertial disturbance of the point load in the force-vibration domain. Similar to Fig. 6a, elastic foundations considerably decrease the amplitude of vibration in both forcedand free-vibration domains.

(a)

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

(b) Fig. 6 Temporal response of the midpoint deflection of an FG plate subjected to: (a) dynamic uniform load and (b) moving point load, for several elastic foundation conditions.

The effect of elastic foundation on the critical buckling load and fundamental frequency of the P-FGM square plates with length-to-thickness ratio (a / h  10) are depicted in Figs. 7a and 7b, respectively. The buckling load and fundamental frequency are obtained by the SCM and TSDT for non-homogeneity index 0 a n a 10 . As shown in Fig. 7, increasing the elastic foundation stiffness remarkably raises the buckling load and fundamental frequency. Furthermore, since increasing the non-homogeneity index n reduces the ceramic volume fraction of the FG plate, the buckling load and fundamental frequency are dropped considerably by increasing n . For instance, increasing n from 0 to 10, in an FG plate rested on the Pasternak-type elastic foundation (k w , k s )  (500,50) , decreases the buckling load and fundamental frequency by 72% and 37%, respectively. Since the non-homogeneity index affects buckling/frequency and weight of FG structures in an inverse manner, a proper value should be chosen for n to satisfy design requirements.

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

(a)

(b) Fig. 7 Effect of elastic foundation condition on (a) critical uniaxial buckling load and (b) fundamental frequency of FG plates for a range of non-homogeneity index 0 a n a 10 .

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A. H. Akbarzadeh, A. Abedini, and Z. T. Chen

6. Concluding Remarks This paper has studied the structural response of an FGM plate resting on a Pasternak-type elastic foundation based on the classical and higher-order shear deformation theories. Several micromechanical models have been employed to obtain the effective material properties of the two-phase FGM plate with the P-FGM, S-FGM, and E-FGM profiles for the volume fraction distribution. The time-dependent governing partial differential equations have been solved by Fourier series expansion and a numerical time integration technique for static, buckling, and free- and forced-vibration analyses of simply-supported FGM plates. Using the semi-analytical method, the effects of explicit micromechanical models, including the Voigt, Reuss, Hashin-Shtrikman bounds, LRVE, and Tamura, as well as the semi-explicit model of SCM on the static and dynamic displacement and stress fields, critical buckling load, and fundamental frequency have been examined. The significane of selecting an appropriate micromechanical model and a proper FGM profile on the accurate estimation of structural properties of FG structures are observed from the numerical results. It is found that the SCM, as the only micromechanical model which satisfies Zuiker’s micromechanical requirements, estimates the structural behavior within the Hashin LB and Hashin UB response domain. The discrepancy between the structural responses of FGMs estimated by the Voigt and other micromechanical models are observed to be noticeably dependant on the non-homogeneity index n . Hashin LB and Hashin UB estimate the maximum deflection of a square FG plate with n  0.5 up to 30% and 12% higher than the Voigt model and the buckling load up to 23% and 11% lower than the Voigt estimation. The numerical results also reveal that the variation of volume fraction by the nonhomogeneity index in the E-FGM is much more remarkable than the P-FGM. For example, the maximum deflection and von Mises stress in an E-FGM plate with n  10 is up to 22% and 4% higher, respectively, than the P-FGM plate. Finally, due to the trade-off between the structural response and the weight of FG structures as a function of non-homogeneity index, FGM profile, and elastic foundation condition, a multi-objective optimization scheme could be employed to reach an optimum design for FGMs.

Appendix A The non-zero components of stiffness and mass matrices as well as the force vector are given here for CLPT. Components of the stiffness matrix are: K11CLPT  A11rm 2 A66 rn 2 , K12CLPT  ( A12  A66 )rm rn , K13CLPT  ( B12  2 B66 )rm rn 2  B11rm 3 CLPT CLPT CLPT K 21  K12CLPT , K 22  A22 rn 2 A66 rm 2 , K 23  ( B12  2 B66 )rm 2 rn  B22 rn 3 , K 31CLPT  K13CLPT CLPT K 32CLPT  K 23

K 33CLPT  ( 2 D12 4 D66 )rm 2 rn 2 D11rm 4 D22 rn 4 KW K S (rm 2  rn 2 ) ( N xx0 rm 2  N yy0 rn 2 ) (A.1)

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The components of the mass matrix are specified as: CLPT CLPT CLPT M 11CLPT  I 0 , M 13CLPT  I1rm , M 22  I 0 , M 23  I1rn , M 31CLPT  M 13 , M 32CLPT  M 23

M 33CLPT  ( I 0  I 2 (rm 2  rn 2 )) (A.2) CLPT 31

The non-zero force vector is: F

 Qmn .

Appendix B The non-zero components of stiffness matrix for FSDT/TSDT are: K11TSDT  ( A11rm 2  A66 rn 2 ) , K12TSDT  ( A12  A66 )rm rn , K13TSDT  c1 E11rm 3  c1 ( E12  2 E66 )rm rn 2

K14TSDT  ( B11 c1 E11 )rm 2 ( B66 c1 E66 )rn 2 , K15TSDT  ( B12  B66 c1 ( E12  E 66 ))rm rn , TSDT TSDT K 21  K12TSDT , K 22  ( A66 rm 2  A22 rn 2 ) , TSDT TSDT K 23  c1 (2 E66  E12 )rm 2 rn  c1 E22 rn 3 K 24  ( B66  B12 c1 ( E12  E66 ))rm rn , TSDT K 25  ( B66 c1 E66 )rm 2 ( B22 c1 E22 )rn 2 TSDT TSDT TSDT K 31  K13TSDT , K 32  K 23

TSDT K 33  ( A55 2c2 D55  c2 2 F55  K s )rm 2 ( A44 2c2 D44  c2 2 F44  K s )rn 2

2c12 ( H12  2 H 66 )rm 2 rn 2 c12 ( H11rm 4  H 22 rn 4 ) KW ( N xx0 rm 2  N yy0 rn 2 ) TSDT K 34  ( A55 2c2 D55  c2 2 F55 )rm  c1 ( F11 c1 H11 )rm 3  c1 (2 F66 2c1 H 66  F12 c1 H12 )rm rn 2 TSDT K 35  ( A44 2c2 D44  c2 2 F44 )rn  c1 ( F22 c1 H 22 )rn 3  c1 (2 F66 2c1 H 66  F12 c1 H12 )rm 2 rn TSDT TSDT TSDT TSDT TSDT K 41  K14TSDT , K 42  K 24  K 34 , K 43 TSDT K 44  ( D11 2c1 F11  c12 H11 )rm 2 ( D66 2c1 F66  c12 H 66 )rn 2  ( A55  2c2 D55 c2 2 F55 ) TSDT TSDT TSDT TSDT , K 45  ( D12 2c1 F12  c12 H12 2c1 F66  c12 H 66  D66 )rm rn , K 51  K15TSDT , K 52  K 25 TSDT TSDT TSDT TSDT , K 54 K 53  K 35  K 45 TSDT K 55  ( D22 2c1 F22  c12 H 22 )rn 2 ( D66 2c1 F66  c12 H 66 )rm 2  ( A44  2c2 D44 c2 2 F44 ) (B.1) The mass matrix components for FSDT/TSDT are: TSDT TSDT TSDT M 11TSDT  I 0 , M 13TSDT  c1 I 3rm , M 14TSDT  J1 , M 22  I 0 , M 23  c1 I 3 rn , M 25  J1 TSDT TSDT TSDT TSDT TSDT M 31  M 13TSDT , M 32  M 23  I 0 c12 I 6 (rm 2  rn 2 ) , M 34  c1 J 4 rm , M 33 TSDT TSDT TSDT TSDT TSDT TSDT TSDT M 35  c1 J 4 rn , M 41  M 14TSDT , M 43  M 34  K 2 , M 52  M 25 , M 44 TSDT TSDT TSDT M 53  M 35  K2 , M 55

(B.2) TSDT 31

The non-zero component of the force matrix is: F

27

 Qmn .

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