Porosity influence on structural behaviour of skew functionally graded magneto-electro-elastic plate

Accepted Manuscript Porosity influence on structural behaviour of skew functionally graded magnetoelectro-elastic plate M.C. Kiran, S.C. Kattimani, M. Vinyas PII: DOI: Reference:

S0263-8223(17)33970-3 https://doi.org/10.1016/j.compstruct.2018.02.023 COST 9373

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

27 November 2017 23 January 2018 12 February 2018

Please cite this article as: Kiran, M.C., Kattimani, S.C., Vinyas, M., Porosity influence on structural behaviour of skew functionally graded magneto-electro-elastic plate, Composite Structures (2018), doi: https://doi.org/10.1016/ j.compstruct.2018.02.023

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 proof before it is published in its final 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.

Porosity influence on structural behaviour of skew functionally graded magneto-electro-elastic plate Kiran M C, S C Kattimani, Vinyas M1 Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal-575025 Abstract: This article presents a finite element (FE) formulation to assess the influence of porosity on the static responses and free vibration of functionally graded skew magnetoelectro-elastic (FGSMEE) plate. The porosity is accounted for local density using modified power law. The skew edges of the plate are achieved by implementing transformation matrix. The coupled constitutive relations establish the different couplings associated with MEE materials. The displacements, potentials, and stresses for the porous skew plate are established through static analysis. The influence of porosity on the natural frequency of the skew plate is investigated via free vibration analysis. The influence of different porosity distributions on various skew angles of the FGSMEE plate has been studied. The effect of porosity volume, skew angle, and geometrical parameters such as aspect ratio, thickness ratio, and boundary conditions on the porous FGSMEE plate is investigated. Keywords: porosity; functionally graded; free vibration; magneto-electro-elastic; skew plate; static analysis.

1

Email Id: [email protected]

1

1. Introduction: The piezoelectric (BaTiO3) and magnetostrictive (CoFe2O4) constituents form the magneto-electro-elastic (MEE) material. Such MEE materials display a unique magnetoelectric effect which is absent in their individual constituents and found to be critical for sensor and actuator applications [1]. Magneto-electro-elastic (MEE) materials are not necessarily functionally graded (FG) and also finds its presence in layered form depending on the requirement of the application [2 – 4]. Functionally graded (FG) materials exhibit enhanced structural characteristics and are gradually expanding their presence in many commercial applications [5, 6]. A number of review articles and historical notes on various plate theories notify the different methods to analyse the structural behaviour of plates [7 – 13]. The basic studies concerning the structural behaviours of functionally graded MEE plates have gained attention of many researchers [14 – 16]. The free vibration characteristics provided the natural frequencies associated with FG MEE plates of several combinations using different methodologies [17 – 20]. The static behaviour of FG MEE plates was well established based on many theories and different solution methods [21 – 23]. Recently, the Carrera unified formulation is used by many researchers to assess the structural characteristics of the MEE plates [24 – 28]. Milazzo [29, 30] proposed unified formulation for layered and functionally graded MEE structures. The buckling behaviour of FG MEE plate resting on a elastic foundation was performed by Li et al [31]. Vinyas and Kattimani [32, 33] analysed the influence of thermal and hygrothermal environments on the static behaviour of functionally graded plates. Kattimani and Ray [34, 35] performed the control of geometric nonlinear vibration of a FG MEE plate using active control layer damping technique. The FG structures possess enhanced structural properties. However, porosities are the major concern which is infused into the material during fabrication. Among the various fabrication processes of the FG materials, the sintering process is found to be the most sought due to its cost-effectiveness [36 – 37]. In sintered FG structures, porosity is the commonly found defects. The FGM prepared using sintering process possesses micro-voids or porosities due to the different solidification rate of material constituents [38 – 41]. A recent study [42] suggests the importance of considering porosity factor in the design and analysis of FGMs. Recently, Ebrahimi et al. [43] analyzed the vibration characteristics of MEE heterogeneous porous material plates resting on elastic foundations. Porosity-dependent vibration analysis of piezo-magnetically actuated heterogeneous nanobeams was performed by Ebrahimi and 2

Barati [44]. Shahverdi and Barati [45] obtained the vibration characteristics of porous FG nanoplates. Shafiei et al. [46] characterized the vibration behaviour of two-dimensional imperfect FG porous nano and micro-beams. Aero-hygro-thermal stability analysis of higherorder refined supersonic FGM panels with even and uneven porosity distributions was studied by Barati and Shahverdi [47]. Using refined four-variable theory, Barati et al. [48] studied the electro-mechanical vibration of smart piezoelectric FG plates with porosities. Ebrahimi et al. [49] studied the free vibration of smart porous plates subjected to various physical fields considering the neutral surface position. The geometrical changes in terms of the plate skewness introduce an effective behavioural change in the FG plate. The studies involving the evaluation of structural characteristics of skew FG plates have been reported. Free vibration of FG quadrilateral microplates in the thermal environment was studied by Shenas and Malekzadeh [50]. Ruan and Wang [51] investigated the transverse vibrations of moving skew plates made of FG material. Adineh and Kadkhodayan [52] carried out three-dimensional thermo-elastic analysis and also obtained the dynamic response of a multi-directional FG skew plate on elastic foundation. Free vibration characteristics of FG-CNT reinforced composite skew plates were assessed by Kiani [53]. Gracia et al. [54] investigated the static and free vibration behaviour of FG-CNT reinforced skew plates. Ardestani et al. [55] developed isogeometric analysis to assess the effect of CNT orientation on the static and vibration response of CNT-reinforced skew composite plates. An analytical investigation of dynamic instability of FG skew plate under periodic axial compression was carried out by Kumar et al. [56]. In spite of tremendous developments in manufacturing techniques, porosity is the common defect most often associated with FG material structures. Looking into the literature, a finite element study of porous functionally graded skew magneto-electro-elastic (FGSMEE) plate to assess the free vibration and static behaviour is unavailable. Hence, the present article intends to establish a suitable FE model for FGSMEE plate accounting the porosity. Such FE formulation is utilized to evaluate the free vibration and static characteristics of porous FGSMEE plate for different skew angles and porosity models. The effect of different skew angle, porosity distribution, porosity volume, and gradient index affecting the structural behaviour of porous FGSMEE plate is extensively investigated. Further, the effect of thickness ratio, aspect ratio, and boundary conditions is also studied. 2. Problem description 2.1 Geometry and constraints

3

The geometry of porous FGSMEE plate is schematically depicted in Fig. 1(a). The length, the width and the total thickness of the plate are a, b and h, respectively. The skew angle of the porous FGSMEE plate is α. Figure 1(b) illustrates the top view of the porous FGSMEE plate. The material properties of the porous FGSMEE plate are assumed to vary across the thickness with the bottom surface of the plate being piezoelectric (BaTiO3) and the top surface being magnetostrictive (CoFe2O4). 2.2 Displacement equation The plate model involved in the present analysis is developed by Hilderbrand et al. [7]. The displacement components u, v and w along x-, y-, and z-direction at any point in the porous FGSMEE plate can be represented by [7, 59] u(x, y, z, t) = u0 ( x, y, t )  z θx ( x, y, t ) v(x, y, z, t) = v0 ( x, y, t )  z θ y ( x, y, t ) w(x, y, z, t) = w0 ( x, y, t )  z θz ( x, y, t )  z 2 z ( x, y, t )

(1)

where, u0 and v0 are the translational displacements along the x- and y- directions while w0 is the transverse displacement along z-direction at any point in the FGSMEE plate. θx and θy correspond to the generalized rotation about the y- and x-axis, respectively. Simultaneously, θz is the generalized rotational displacements for the FGSMEE plate with respect to the thickness coordinate and  z is the higher order terms of the Taylor series of expansion. The boundary conditions of the plate are considered as follows: (a)

Simply supported edges

at x = y tanα, x = a + y tanα :

v01 = w01 =  y1 =  z1 =  1 =  1 = 0

at y = 0, y = b cosα:

u0 = w0 =  z =  =  = 0

(b)

Clamped edges

at x = y tanα, x = a + y tanα:

u01 = v01 = w01 =  x1 =  y1 =  z1 =  1 =  1 = 0

at y = 0, y = b cosα:

u0 = v0 = w0 = x =  y =  z =  =  = 0

(c)

(3)

Free edges

at x = y tanα, x = a + y tanα:

u01 = v01 = w01 =  x1 =  y1 =  z1 =  1 =  1  0

at y = 0, y = b cosα:

u0 = v0 = w0 = x =  y =  z =  =   0

where,

(2)

(4)

and ψ are the electric and magnetic potential degrees of freedom as specified in

Appendix-1. Also, the superscripts appearing in Eqs. (2) – (4) correspond to the degrees of

4

freedom on the skew edges of the plate.

For the ease of computation, rotational and

translational displacements are considered separately as follows:  u0 v0 w0  , d r   θx θ y θz  z  2.3 Coupled constitutive equation

dt 

T

T

(5)

Considering the effect of coupled fields, the constitutive equations for the FGSMEE plate can be expressed as follows [35]:

 b   [Cb ( z)]{ b }  {eb ( z)}Ez  {qb ( z)}H z , {σ s }  [Cs ( z )]{ s }

(6a)

Dz  {eb ( z )} { b }  33 ( z) Ez  d33 ( z) H z

(6b)

Bz  {qb ( z )} { b }  d33 ( z ) Ez  33 ( z ) H z

(6c) and 33 are the

T

T

k where, Cbk  and Csk  are the transformed coefficient matrices [60, 61],  33

dielectric constant and the magnetic permeability constant, respectively; d33 is the electromagnetic coefficient. Since the plate is considered to be thin, the electric displacement, the electric field, the magnetic induction, and the magnetic field along the z-direction are only considered and represented by Dz, Ez, Bz, and Hz, respectively. The matrices corresponding to Eq. (6) are provided in Appendix-1. 2.4 Porosity inclusion using modified power law The coefficients are bound by power-law distribution given as follows [43]: Pfg ( z)  PF  ( PB - PF ) V  ( PB + PF )  (m / 2) V p

(7)

where, Pfg is the generalized term to represent the material property variables such as C ,  , e , q ,  , and  . The subscripts B and F refer to BaTiO3 and CoFe2O4, respectively. Vp is the

generalized term to represent the different porosity distributions (i.e., Vu, Vo, Vx, Vv) while, m is the porosity index (0 < m < 1) and V is given by  z   1   V=        h   2  

η

(8)

wherein, η is the power law gradient and different porosity distributions shown in Fig. (2) are given by (a) Uniform porosity distribution, (9.a)

Vu =1

(b) Centralized porosity distribution,   2z Vo = 1  h  

    

(9.b)

(c) High density of porosity at the top and the bottom of the plate while low at the midspan i.e., 5

 2 z Vx =   h

  

(9.c)

(d) High density of porosity at the top and low at the bottom,  2 z Vv = 1  h 

  

(9.d)

2.5 Equation of motion Employing the principle of virtual work [34], the governing equations for the porous FGSMEE plate can be established as   T    { b }{σ b }d     { s }{σ s }d     dt   ( z ) dt d      Ez Dz d     H z Bz d       

 

   dt  Ft dAel  0

(10)

T

A

where,  indicates the volume of the plate, Ft is the applied force with sinusoidal distribution on the top surface area Ael, ρ(z) denotes the mass density variation through the thickness. According to Maxwell’s equation, the transverse electric field (Ez) and the transverse magnetic field (Hz) is given by

Ez  

  and H z   z z

(11)

2.6 Finite element formulation and equations of motion The FGSMEE plate is discretized using eight noded iso-parametric elements. At any point within the element, the generalized displacement vectors dt  and d r  , the magnetic potential vector { } and the electric potential vector {} can be expressed in terms of nodal

 

generalized displacement vectors {dtel } and {d rel } , the nodal magnetic potential vector  el

   d    N d  , d    N d  ,  =  N    and  =  N   

and the nodal electric potential vector  el , respectively, as follows: t



and

el t

t

r

el r

r

el



el

Nti  ni It , Nri  ni I r

(12) (13)

where,  Nt  ,  N r  ,  N  and  N  are the shape function matrices [60, 61], It and Ir are the identity matrices, and ni is the shape function of natural coordinate associated with the ith node. Considering the shape functions, Eq. (11) can be modified as follows:

 

 

1 1 Ez    N   el and H z    N   el h h 6

(14)

Now, the generalized strain vectors at any point within the element can be expressed in terms of the nodal generalized strain vectors as follows:

 tb   btb dtel  ,  rb   brb drel   ts   bts dtel  ,  rs   brs drel 

(15)

in which, btb  , brb  , bts  and brs  are the nodal strain-displacement matrices. Substituting Eqs. (6), (13), (14) and (15) into Eq. (10) and simplifying, we obtain the elemental equations of motion for the FGSMEE plate as follows:

 

           d    k  d    k      k     0  k  d   k  d   k     0  k  d   k  d    k     0

 M el  dtel  kttel  dtel  ktrel  drel  ktel   el   ktel   el  Ft el el T tr

 k

el t

el t

el t

el rr

T

el r

el t

T

el r

el t

el r

el r

T

T

el

el r

el r



el

el r



el

el

el

el

The matrices and the vectors appearing in Eqs. (16) – (19) are provided in Appendix-2.

(16) elem (17) ental (18) equat (19) ions for )

2.7 Skew boundary transformation

In case of FGSMEE plates, the supported adjacent edges of the boundary element are not parallel to the global axes (x, y, z). Hence, in order to specify the boundary conditions at the skew edges of the plate, the displacements u’, v’ and w’ at any point on the skew edges of the local coordinates must be restrained along the x’-, y’- and z’-directions. The boundary conditions can be specified conveniently by transforming the element matrices corresponding to the global axis to the local axis along the edges. A simple transformation relation can be expressed between the local degrees of freedom and the global degrees of freedom for the generalized displacement vectors of a point lying on the skew edges of the plate as follows [57, 58]:

dt    Lt dt'  , dr    Lr dr'  (20) d   u v w  , d   θ θ θ   d  , d  are the displacements on the global and the local edge ' t

where, dt  , d r  and

' t

' 0

' 0

' T 0

' r

' x

' y

' z

' T z

' r

coordinate system, respectively.  Lt  and  Lr  are the transformation matrices [57, 58] for a node on the skew boundary and is given by  c s 0 0  c s 0  s c 0 0   Lt    s c 0 ,  Lr    0 0 1 0  0 0 1     0 0 0 1

7

(21)

in which, c  cos  and s  sin  , the skew angle of the plate is α. It may be noted that for the nodes which do not lie on the skew edges, the transformation from global coordinates to the local coordinates is not required. The transformation matrices in such cases are the diagonal matrices in which the values of the principal diagonal elements are unity. The elemental stiffness matrices of the element containing the nodes laying on the skew edges are given as follows:  k ttel   T T  k el  T  ,  k trel   T T  k el  T  , 1 1  tt  1    tr  2    k elrr   T T k el  T  ,  M el   T T  M el  T  1 2    1  rr  2    where, the transformation matrices [T1] and [T2] are given by [57, 58]  Lt    o  o  o T1    o   o  o   o

o

 Lt  o o o o o o

o o  Lt  o o o o o

o o o  Lt  o o o o

o o o o  Lt  o o o

o o o o o L  t o o

o o o o o o  Lt  o

o   o  o   o  o   o  o    Lt 

,

 Lr    o  o  o T2    o   o  o   o

o

 Lr  o o o o o o

o o  Lr  o o o o o

o o o  Lr  o o o o

o o o o  Lr  o o o

(22)

o o o o o L  r o o

o o o o o o Lr  o

o   o  o   o  o   o  o   Lr 

(23)

in which, o and o are the (3×3) and (4×4) null matrices respectively, and the number of  Lt  and  Lr  matrices are equal to the number of nodes in the element. The elemental equations of motion are assembled to obtain the global equations of motion of the FGSMEE plate as follows:

 M dt   kttg  dt   ktrg  dr   ktg    ktg     Ft   ktrg 

T

(24)

dt   krrg  dr   krg   krg     0

(25)

dt   krg  dr   kg    0

(26)

ktg 

T

T

T T g (27) ktg  dt   krg  dr   k     0 The matrices appearing in Eqs. (24) – (27) are defined in Appendix-2. Solving the global

equations of motion (Eqs. (25) - (27)) to obtain global generalized displacement vector dt  and d r  by condensing the global degrees of freedom for   and   in terms of d r  as follows:

   kg 

1

1

1

T

g ktg  dt  k  krg  1

T

dr  ,

  kg  ktg  dt   kg  krg  dr  , 1 T dr     K3   K2  dt  T

8

T

(28)

Now, substituting Eq. (28) in Eq. (24) and upon simplification, we obtain the global equations of motion in terms of the global translational degrees of freedom as follows:

 M dt    K dt   Ft 



 K2    K1  ,  K 2  ,

and  K    K1    K 2  K3  where,

 K  is

1

the equivalent stiffness matrix and

(29) T

and

 K3  are

the global

aggrandized matrices and are provided in Appendix-2. 3. Results and discussion 3.1 Validation studies The proposed FE model of FGSMEE plate is verified by comparing the results with the studies available in the literature [30, 58, and 59]. Milazzo [30] implemented equivalent single layer theory using a 9 noded quadrilateral element with 8 × 8 mesh size and Garg et al. [59] presented the free vibration behaviour skew plates using the higher order shear deformation theory by discretising the plate using a 6 × 6 mesh of 16 noded element. The normalized natural frequencies for the rectangular simply supported FGMEE plate with an aspect ratio of b/a = 2 and thickness ratio of h/a = 0.1 and 0.2 are presented in Table 2. Considering the convergence criteria, the results are obtained for various element mesh size. It can be clearly seen from the tabulated results that for a 20 × 20 mesh size, an excellent agreement is achieved with the solutions available in Milazzo [30]. Therefore, for all the subsequent analysis, a mesh size of 20 × 20 is considered. It may be noted that to the best of authors’ knowledge, the study related to FGSMEE plates are not available in the literature. Hence, the effectiveness of the FE formulation to assess the influence of skew angle on the plate characteristics is evaluated for the laminated composite plate by degenerating the coupling coefficients of the FGSMEE, and the results are presented in Table 3 and 4. The tabulated results display an excellent agreement with the results reported in the literature [58] and [59]. Hence, the correctness of the present FE formulation is assessed and further extended the procedure to evaluate the structural characteristics of porous FGSMEE plate. 3.2 Free vibration assessment of porous FGSMEE plate Several parameters influencing the free vibration characteristics of porous FGSMEE plate is discussed in this section. Different skew angles, porosity distributions, material gradient index, and porosity volume influencing the natural frequency of the plate are explicitly investigated. In addition, the influence of geometric parameters such as thickness ratio, aspect ratio, and boundary condition on the natural frequency is also studied. The 9

geometrical parameters considered for the study are given as follows: a/h = 100 and b/a = 1. Tables 5 – 8 display the influence of skew angle for different porosity distributions on the natural frequency of porous FGMEE plate. The boundary condition considered for analysis is simply supported on all the edges and the porosity index is taken to be 0.1. It can be observed from these tables that the increase in skew angle increases the natural frequency of the plate. However, the natural frequency drastically increases for α = 450. In addition, these tables suggest that the inclusion of porosity brings down the natural frequency in comparison with that of perfect FG plate. Further, every porosity distributions display a unique influence on the natural frequency with Vu distribution exhibiting the lowest natural frequency while Vo obtains the highest. The increase in skew angle causes the reduction in the plate area and imparts additional stiffness causing the increase in natural frequency. In addition, the inclusion of porosity brings down the stiffness of the material and decreases the natural frequency. The material gradient index affecting the free vibration characteristics of porous FGSMEE plate is presented in Figs. 3(a) – (d). These figures suggest that the increase in gradient-index lowers the natural frequency for all the skew angles and porosity distributions. It is interesting to note that for Vo distribution at α = 450, the increase in gradient-index meagrely affects the natural frequencies. Further, the influence of volume of porosity on the free vibration behaviour is presented in Fig. (4). It can be seen that the increase in the porosity volume decreases the natural frequency significantly. It can also be noted that the influence of the increase in porosity volume on Vo distribution is minimal in comparison with other porosity distributions. The influence of geometric parameters on the free vibration behaviour of the porous FGSMEE plate is investigated. The analysis is performed for the plate with α = 450. Table 9 presents the effect of thickness ratio on the natural frequency of porous FGSMEE plate. The increase in the thickness ratio increases the natural frequency of the plate. The thick and moderately thick plate display a rapid increase in the natural frequency while the thin plate (a/h ≥ 50) witness a steady increase in the natural frequency. Simultaneously, the effect of aspect ratio on the free vibration characteristics of FGSMEE plate is analysed and the results are tabulated in Table 10. The increase in aspect ratio decreases the natural frequency of the plate. It is also observed that for the higher aspect ratio, the natural frequencies decreases steadily while decreases rapidly for lower aspect ratio. 3.3 Static characteristics of porous FGSMEE plate The behaviour of the porous FGSMEE plate subjected to static load is studied in this section. The plate with a/h = 100, b/a = 1 and m = 0.1 is considered for the analysis. The 10

effect of skew angle on the various porosity distributions is investigated. A comparison study of the porous and non-porous plate is carried out to highlight the influence of porosity on the structural behaviour of the skew plate. In addition, the geometric parameters influencing the structural behaviour are also assessed. The effect of skew angle and porosity distribution is depicted in Figs. (5) – (14). Figures 5 (a) – (c) present u displacement for three different skew angles. A distinctive behaviour is evident for every porosity distribution with the largest displacement being associated with Vv distribution and the lowest with Vo. However, Vu and Vx distributions display an identical behaviour. It is interesting to note that the increase in skew angle results in lower u displacement and the lowest displacement is witnessed for α = 450. The v displacement characteristics of the plate are shown in Figs. 6 (a) – (c). It can be observed from these figures that the increase in skew angle increases the stretching and reduces the bending in the porous FGSMEE plate. It may also be observed from these figures that the v displacement display combined stretching and bending for α = 150. However, for α = 450 stretching contribute majorly to the v displacement. In addition, irrespective of skew angle, every porosity distribution produced unique behaviour with the largest displacement being seen for Vv distribution and the lowest with Vo. The electric potential

presented in

Figs. 7 (a) – (c) decreases with the increase in skew angle. These figures also suggest that the porosity distributions display significant influence on the electric potential. The magnitude of the characteristic curve of electric potential for Vu and Vx distribution is nearly identical. However, Vo distribution registers the lowest electric potential. The increase in skew angle decreases the magnetic potential as shown in Figs. 8 (a) – (c). Among the porosity distributions considered, Vv registers the largest magnetic potential while for the other distributions display nearly identical behaviour. The stress components presented in Figs. (9) – (12) decreases with the increase in skew angle. It can also be seen that the porosity distributions have a minor influence on the stress magnitudes. Figures 13 (a) – (c) suggest that the magnetic induction significantly increases with the increase in skew angle. It is also interesting to note that Vv distribution registers the largest magnetic induction while the other distributions attain nearly identical magnitudes. A significant influence of porosity distributions on the electric displacement is seen from Figs. 14 (a) – (c). The increase in skew angle increases the electric displacement. Meanwhile, the increase in skew angle nullified the influence of porosity distribution on the electric displacement. A comparison study of porous and nonporous FGSMEE plate subjected to static load is performed. The skew angle for the plate is taken to be 45 0 and rest of the geometrical conditions are considered similar to previous analysis. Figures (15) – (24) present the 11

comparison study of porous and non-porous plate via primary quantities (u, v, , and ψ) and secondary quantities (stresses, magnetic induction, and electric displacement) for four different porosity distributions. Displacements, electric potential, and electric displacement are higher for Vu, Vx, and Vv porous plate over the non-porous plate while for Vo distribution a meager influence is seen. The magnetic potential increases for Vv distribution while Vu, Vo, and Vx porous plate produce no major influence in comparison with the non-porous plate. The influence of porosity on the stresses is negligible. Magnetic induction is higher for nonporous plate than the porous plate for Vu, Vo, and Vv while for the Vx porous plate, major influence has not been seen. The geometrical parameters influencing the static behaviour of porous FGSMEE plate is presented in Figs. (25) – (28). The primary quantities (u, v,

, and ψ) and the secondary

quantities (stresses, magnetic induction, and electric displacement) investigated for different aspect ratios are presented in Figs. 25 (a) – (j). These figures suggest that the increase in aspect ratio increases the primary and secondary quantities. Further, increase in lateral dimensions increases the influence of porosity on FGSMEE plate. Similarly, the static characteristics of porous FGMEE plates are studied for different thickness ratios. Figures 26 (a) – (j) suggest that the porosity largely affects the thin plate over thick plates. In addition, boundary conditions affecting the plate characteristics for static load are presented in Figs. 27 (a) – (j). It can be seen from these figures that the simply supported plate produces larger displacements, potentials, stress, magnetic induction, and electric displacement over clamped plates. The influence of volume of porosity on the static behaviour of porous FGSMEE plate is also assessed. The study has been carried out for different porosity distributions as shown in Figs. (28) – (37). It can be noticed that the higher porosity volume results in higher displacements, potentials, stresses, magnetic induction, and electric displacement for Vu, Vx, and Vv porosity distributions. However, the influence of porosity volume on Vo porosity distribution is found to be ineffective for porosity volume, m > 0.2. It can also be noticed from the above figures that the higher porosity volume effectively brings down the stiffness of the plate thereby results in higher primary and secondary quantities. The negligible influence of higher porosity volume (i.e., m > 0.2) on Vo suggests that the existence of porosity near to the bottom and top surface are more prone to higher displacement and stresses over porosity distributions near to mid-plane. Conclusions The free vibration and static response of porous FGSMEE plate are studied using a finite element formulation. The porosity is accounted for local density and implemented 12

using the modified power law. The skew edges of the plate are achieved by using transformation matrix. The influence of skew angle and porosity on the natural frequency of the FGMEE plate has been studied. The natural frequencies of the plate increase with increase in skew angle and a drastic rise is seen for α = 450. The inclusion of porosity brings down the natural frequency in comparison with non-porous plates. The porosity distribution pattern Vo exhibit the least influence on the natural frequency while Vu displays the largest effect for all the skew angle of the FGSMEE plate. In addition, the increase in gradient index and porosity volume meagrely affects Vo distribution skew plate. In general, for the FGSMEE plate, the presence of larger porosity density near to its top and bottom surface significantly influence the free vibration behaviour of the plate. Further, it is also observed that the increase in thickness ratio increases the natural frequency while the increase in aspect ratio decreases the natural frequency. Static studies reveal that the increase in skew angle decreases the u-displacement and the porosity distribution Vv exhibits the largest displacement while the Vo has the lowest. The increase in skew angle increases the stretching and decreases the bending for v-displacement. The electric potential decreases with the increase in skew angle and the lowest electric potential are noticed for Vo porosity pattern. Moreover, the magnetic potential decreases with the increase in skew angle and the porosity pattern Vv registers lowest magnetic potential. It is also observed that the porosity displays a negligible influence on the stresses, whereas the stress magnitude decreases with the increase in skew angle. The magnetic induction and electric displacement increase with the increase in skew angle. A comparison study of porous and non-porous plate display the noticeable influence of porosity on the plate structural characteristics. Higher porosity volume results in higher displacements, potentials, stresses, magnetic induction and electric displacement for Vu, Vx, and Vv porous skew plate while higher porosity volume displays negligible influence on Vo porous skew plate. Appendix-1 The FG material coefficient matrices, the electric coefficient matrix, and the magnetic coefficient matrix appearing in Eq. 6 (a) – (c) are given as follows [34]:

 C11 ( z )  C ( z) Cb ( z )    12 C13 ( z )  C16 ( z )

C12 ( z ) C22 ( z ) C23 ( z ) C26 ( z )

C13 ( z ) C23 ( z ) C33 ( z ) C36 ( z )

C16 ( z )   C26 ( z )  C ( z ) C45 ( z ) , Cs ( z )    55  C36 ( z )  C45 ( z ) C44 ( z )   C66 ( z ) 

13

e31 ( z )  q31 ( z )  e ( z )  q ( z )   32   32  , {eb ( z )}   { q ( z )}     b e33 ( z )  q33 ( z )  e36 ( z )  q36 ( z ) 

(a)

The stress matrices i.e., σ b  and σ s  appearing in Eq. (10) are given by

σb 

 σ x σ y σ z σ xy  , σ s    xz  yz  T

T

(b)

The stresses are derived from strains given by

 b    x  y  z  xy  ,  s    xz  yz  T

T

also,  b    tb    Z1  rb  ,  s    ts    Z 2  rs 

 u { tb }   0  x

(c)

 w w0  v0 u0 v0  0  , { ts }   0   and y y x   x y 

  { rb }   x  x

 y

  x v0  θz  z  x x 

y

(d)

wherein, the transformation matrices [Z1] and [Z2] are expressed as z 0 Z =  1  0  0

0 z 0 0

0 0 0 z

0 0 0 0  ,  Z 2 = 1 2z   0 0

1 0 z 0 z 2  0 1 0 z 0

0  z2 

where,

σ x , σ y and σ z

= Normal stresses along x-, y- and z-directions

σ xy

= In-plane shear stress

 xz and  yz

= Transverse shear stresses along xz- and yz- directions

Cb ( z )  and Cs ( z )  33 ( z ) and 33 ( z )

= Functionally graded material coefficient matrices

d33 ( z )

= The electromagnetic coefficient.

Dz Ez Bz Hz

= electric displacement = Electric field = Magnetic induction = Magnetic field = The electric coefficient matrix

{eb ( z )}

= dielectric constant and the magnetic permeability constant

qb ( z)

= The magnetic coefficient matrix



= The electric potential.

14

(e)



= The magnetic potential

Appendix-2 The elemental matrices and vectors appearing in Eqs. (16) – (19) are given by el el el el kttel    ktbel    ktsel  , ktrel   krbel   ktrs  , krr   krrb   k rrs  ,

 ktel    kelt  T ,  ktel    kelt  T , ael bel

 ktel   

ael bel

el  btb   Dt   N  dx dy , kr    T

0 0

 k   

 b 

T

tb

ael bel

 Dt   N  dx dy ,  k    el r

0 0

 k   

 Dr   N  dx dy ,

 b 

T

rb

 Dr   N  dx dy ,

0 0

ael bel

el

T

rb

0 0

ael bel

el t

 b 

  N 

T

ael bel

 D   N  dx dy ,  k    el

0 0

  N 

T

 D   N  dx dy (f)

0 0

where,  Dt  ,  Dr  ,  Dt  ,  Dr  ,  D  and  D  are the rigidity matrices appearing in Eq.(f) are given as follows: h /2

h /2

1 1  Dt    eb ( z ) dz ,  Dt    qb ( z ) dz , h h  h /2  h /2  Dr  

h /2

h/ 2

1 1 T T  Z1  eb ( z ) dz ,  Dr     Z1  qb ( z ) dz ,  h h  h /2 h / 2  D  

33 ( z ) 1 0 h

1 0 1  ,  D   h 33 ( z )  

(g)

The global aggrandized matrices appearing in Eq. (29) are given by 1

g  ktg    ktg   k   ktg  ,

 K2   ktrg   ktg  kg 

1

g  krg    ktg   k   krg  ,

 K3   krrg   krg  kg 

1

g  krg    krg   k   krg  .

 K1   kttg   ktg  kg 

T

1

T

1

T

T

1

T

T

where,  M el 

= Elemental mass matrix

 kttel  ,  ktrel  and  krrel 

= Elemental elastic stiffness matrices

 ktel  ,  krel 

= the elemental electro-elastic coupling stiffness matrices

15

(h)

 ktel  ,  krel 

= the elemental magneto-elastic coupling stiffness matrices

 kel 

= the elemental electric stiffness matrices

el  k 

= the elemental magnetic stiffness matrices

F  M  el

t

= the elemental mechanical load vector = the global mass matrix

 kttg  ,  ktrg  and  krrg 

= the global elastic stiffness matrices

 ktg  and  krg 

= the global electro-elastic coupling stiffness matrices

 ktg  and  krg 

= the global magneto-elastic coupling stiffness matrices

 kg 

= the global electric stiffness matrices

g  k 

= the global magnetic stiffness matrices

Ft 

= the global mechanical load vector

16

References [1] Boomgaard VJ, Born RA, Sintered magnetoelectric composite material BaTiO3-Ni (Co, Mn)Fe2O4. J. Mater. Sci. 1978; 13(7): 1538-48. [2] Buchanan GR, Layered versus multiphase magneto-electro-elastic composites Compos. Part B (Engg) 2004; 35(5): 413–420. [3] Pan E, Han F, Exact solutions for functionally graded and layered magneto-electro-elastic plates. Int. J. Eng. Sci. 2005; 43: 321-339. [4] Kiran MC, Kattimani SC, Buckling characteristics and static studies of multilayered magneto-electro-elastic plate. Struct. Eng. Mech 2017; 64(6): 751-763. [5] Swaminathan K, Naveenkumar DT, Zenkour AM and Carrera E, Stress, vibration and buckling analyses of FGM plates—a state-of-the-art review. Compos. Struct. 2015; 120: 1031. [6] Thai HT, Kim SE, A review of theories for the modeling and analysis of functionally graded plates and shells. Compos. Struct. 2015; 128: 70-86. [7] Hildebrand FB, Reissner E, Thomas GB, Notes on the foundations of the theory of small displacements of orthotropic shells. NACA Technical Note, 1949; No. 1833. [8] Reissner E, Reflections on the theory of elastic plates. Appl. Mech. Rev. 1985; 38(11): 1453-1464. [9] Caliri MF, Ferreira AJ, Tita V, A review on plate and shell theories for laminated and sandwich structures highlighting the Finite Element Method. Compos. Struct. 2016; 156: 6377. [10] Abrate S, Di Sciuva M, Equivalent single layer theories for composite and sandwich structures: A review. Compos. Struct. 2017; 179: 482-494. [11] Thai HT, Kim SE, A review of theories for the modeling and analysis of functionally graded plates and shells. Compos. Struct. 2015, 128: 70-86. [12] Carrera E, Historical review of zig-zag theories for multilayered plates and shells. Appl. Mech. Rev. 2003; 56(3): 287-308. [13] Saravanos DA, Heyliger PR, Mechanics and computational models for laminated piezoelectric beams, plates, and shells. Appl. Mech. Rev. 1999; 52: 305-320. [14] Xiao D, Han Q, Liu Y, Li C, Guided wave propagation in an infinite functionally graded magneto-electro-elastic plate by the Chebyshev spectral element method. Compos. Struct. 2016; 153: 704-711. [15] Hosseini M, Mofidi MR, Jamalpoor A, Jahanshahi MS, Nanoscale mass nanosensor based on the vibration analysis of embedded magneto-electro-elastic nanoplate made of FGMs via nonlocal Mindlin plate theory. Microsyst. Technol. 2017; 1-22. [16] Lezgy-Nazargah M, Cheraghi N, An exact Peano Series solution for bending analysis of imperfect layered functionally graded neutral magneto-electro-elastic plates resting on elastic foundations. Mech. Adv. Mater. Struc. 2017; 24(3): 183-199. [17] Ebrahimi F, Barati MR, Magnetic field effects on dynamic behavior of inhomogeneous thermo-piezo-electrically actuated nanoplates. J. Braz. Soc. Mech. Sci. & Eng 2017; 39(6): 2203-2223. [18] Milazzo A, Refined equivalent single layer formulations and finite elements for smart laminates free vibrations. Compos. Part B (Engg) 2014; 61: 238-253. 17

[19] Shijie Z, Yan F, Hongtao W, Free vibration studies of functionally graded magnetoelectro-elastic plates/shells by using solid-shell elements. J Vibroeng 2012; 14(4): 903. [20] Wu CP, Lu YC, A modified Pagano method for the 3D dynamic responses of functionally graded magneto-electro-elastic plates. Compos. Struct. 2009; 90(3): 363-372. [21] Wu CP, Chen SJ Chiu KH, Three-dimensional static behavior of functionally graded magneto-electro-elastic plates using the modified Pagano method. Mech. Res. Comm. 2010; 37(1): 54-60. [22] Wu CP, Chiu K, Wang YM, A meshfree DRK-based collocation method for the coupled analysis of functionally graded magneto-electro-elastic shells and plates. Comput Model Eng Sci 2008; 35: 181-214. [23] Haitao D, Wei C Mingzhi L, Static/dynamic analysis of functionally graded and layered magneto-electro-elastic plate/pipe under Hamiltonian system. Chin. J. Aeronaut. 2008; 21(1) 35-42. [24] Carrera E, Cinefra M, Li G, Refined finite element solutions for anisotropic laminated plates. Compos. Struct. 2017; 183: 63-76. [25] Carrera E, Valvano S, Analysis of laminated composite structures with embedded piezoelectric sheets by variable kinematic shell elements. J. Intell. Material Syst. Struct. 2017; 28: 2959-2987. [26] Cinefra M, Carrera E, Lamberti A, Petrolo M, Best theory diagrams for multilayered plates considering multifield analysis. J. Intell. Material Syst. Struct. 2017; 28(16): 21842205. [27] Zappino E, Li G, Pagani A, Carrera E, Global-local analysis of laminated plates by node-dependent kinematic finite elements with variable ESL/LW capabilities. Compos. Struct. 2017; 172: 1-14. [28] Kumar SK, Harursampath D, Carrera E, Cinefra M, Valvano S, Modal analysis of delaminated plates and shells using Carrera Unified Formulation–MITC9 shell element. Mech. Adv. Mater. Struct. 2017, 1-17. [29] Milazzo A, Unified formulation for a family of advanced finite elements for smart multilayered plates. Mech. Adv. Mater. Struc. 2016; 23(9): 971-980. [30] Milazzo A, Refined equivalent single layer formulations and finite elements for smart laminates free vibrations. Compos. Part-B (Engg) 2014a; 61: 238-253. [31] Li YS, Buckling analysis of magnetoelectroelastic plate resting on Pasternak elastic foundation. Mech. Res. Commun. 2014; 56: 104-114. [32] Vinyas M, Kattimani SC, Hygrothermal analysis of magneto-electro-elastic plate using 3D finite element analysis. Compos. Struct. 2017; 180: 617-637. [33] Vinyas M, Kattimani SC, Static analysis of stepped functionally graded magnetoelectro-elastic plates in thermal environment: A finite element study. Compos. Struct. 2017; 178: 63-86. [34] Kattimani SC Ray MC, Control of geometrically nonlinear vibrations of functionally graded magneto-electro-elastic plates. Int. J. Mech. Sci. 2015; 99: 154-167. [35] Kattimani SC, Geometrically nonlinear vibration analysis of multiferroic composite plates and shells. Compos. Struct. 2017; 163: 185-194. [36] Khor KA, Gu YW, Effects of residual stress on the performance of plasma sprayed functionally graded ZrO2/NiCoCrAlY coatings. Mater. Sci. Eng. A 2000; 277(1): 64-76. 18

[37] Seifried S, Winterer M, Hahn H, Nanocrystalline gradient films through chemical vapor synthesis. Scr. Mater. 2001; 44(8): 2165-2168. [38] Watanabe Y, Eryu H, Matsuura K, Evaluation of three-dimensional orientation of Al 3 Ti platelet in Al-based functionally graded materials fabricated by a centrifugal casting technique. Acta Mater. 2001; 49(5): 775-783. [39] Song C, Xu Z, Li J, Structure of in situ Al/Si functionally graded materials by electromagnetic separation method. Mater. Des. 2007; 28(3): 1012-1015. [40] Peng X, Yan M, Shi W, A new approach for the preparation of functionally graded materials via slip casting in a gradient magnetic field. Scr. Mater. 2007; 56(10): 907-909. [41] Zhu J, Lai Z, Yin Z, Jeon J, Lee S, Fabrication of ZrO 2–NiCr functionally graded material by powder metallurgy. Mater. Chem. Phy. 2001; 68(1): 130-135. [42] Wattanasakulpong N, Prusty BG, Kelly DW Hoffman M, Free vibration analysis of layered functionally graded beams with experimental validation. Mater. Des. 2012; 36: 182190. [43] Ebrahimi F, Jafari A, Barati MR, Vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundations. Thin-Wall. Struct. 2017; 119: 33-46. [44] Ebrahimi F, Barati MR, Porosity-dependent vibration analysis of piezo-magnetically actuated heterogeneous nanobeams. Mech. Syst. Sig. Pr. 2017; 93: 445-459. [45] Shahverdi H, Barati MR, Vibration analysis of porous functionally graded nanoplates. Int. J. Eng. Sci. 2017; 120: 82-99. [46] Shafiei N, Mirjavadi SS, MohaselAfshari B, Rabby S, Kazemi M, Vibration of twodimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput. Method Appl. Mech. Eng. 2017; 322: 615-632. [47] Barati MR, Shahverdi H, Aero-hygro-thermal stability analysis of higher-order refined supersonic FGM panels with even and uneven porosity distributions. J. Fluids Struct. 2017; 73: 125-136. [48] Barati MR, Shahverdi H, Zenkour AM, Electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory. Mech. Adv. Mater. Struct. 2017; 24(12): 987-998. [49] Ebrahimi F, Jafari A, Barati MR, Free vibration analysis of smart porous plates subjected to various physical fields considering neutral surface position. Arab. J. Sci. Eng. 2017; 42(5):1865-1881. [50] Shenas AG, Malekzadeh P, Free vibration of functionally graded quadrilateral microplates in thermal environment. Thin-Wall. Struct. 2016; 106: 294-315. [51] Ruan M, Wang ZM, Transverse vibrations of moving skew plates made of functionally graded material. J. Vib. Cont. 2016; 22(16): 3504-3517. [52] Adineh M, Kadkhodayan M, Three-dimensional thermo-elastic analysis and dynamic response of a multi-directional functionally graded skew plate on elastic foundation. Compos. Part B: Engg. 2017; 125: 227-240. [53] Kiani Y, Free vibration of FG-CNT reinforced composite skew plates. Aero. Sci. Tech. 2016; 58: 178-188.

19

[54] García-Macías E, Castro-Triguero R, Flores EIS, Friswell MI, Gallego R, Static and free vibration analysis of functionally graded carbon nanotube reinforced skew plates. Compos. Struct. 2016; 140: 473-490. [55] Ardestani MM, Zhang LW, Liew KM, Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates. Comp. Meth. Appl. Mech. Engg. 2017; 317: 341-379. [56] Kumar R, Mondal S, Guchhait S, Jamatia R, Analytical Approach for Dynamic Instability Analysis of Functionally Graded Skew Plate under Periodic Axial Compression. Int. J. Mech. Sci. 2017; 130: 41-51. [57] Kanasogi RM, Ray M C, Active constrained layer damping of smart skew laminated composite plates using 1-3 piezoelectric composites, J. Compos. 2013, Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2013/824163 [58] Garg AK, Khare RK, Kant T, Free vibration of skew fiber-reinforced composite and sandwich laminates using a shear deformable finite element model. J. Sand. Struct. Mater. 2006; 8: 33-53. [59] Tessler A, An improved plate theory of {1, 2}-order for thick composite laminates. Int. J. Sol. Struct. 1993; 30(7): 981-1000. [60] Reddy JN Mechanics of Laminated Composite Plates and Shells Theory and Analysis. 2nd edition, CRC Press [61] Zienkiewickz OC and Taylor RL, The Finite Element Method, 5th Edition, ButterworthHeinemann, Elsevier Science, Oxford, 1989.

20

(a)

(b) Fig. 1: (a) Functionally graded skew MEE plate. (b) Top view of FGSMEE plate

Fig. 2: Porosity distribution (a) Vu (b) Vo (c) Vx (d) Vv

21

(a)

(b)

(c) (d) Fig 3: Effect of gradient index on natural frequency of porous plate (a) Vu (b) Vo (c) Vx (d) Vv

Fig 4: Effect of porosity volume on natural frequency of porous plate

150

300

22

450 Fig. 5 Effect of porosity distribution on porous FGSMEE plate for u (η = 2)

150

300

450 Fig. 6 Effect of porosity distribution on porous FGSMEE plate for v (η = 2)

150

300

23

450 Fig. 7 Effect of porosity distribution on porous FGSMEE plate for

150

(η = 2)

300

450 Fig. 8 Effect of porosity distribution on porous FGSMEE plate for ψ (η = 2)

150

300

24

450 Fig. 9 Effect of porosity distribution on porous FGSMEE plate for σxx (η = 2)

150

300

450 Fig. 10 Effect of porosity distribution on porous FGSMEE plate for σyy (η = 2)

150

300

25

450 Fig. 11 Effect of porosity distribution on porous FGSMEE plate for σxy (η = 2)

150

300

450 Fig. 12 Effect of porosity distribution on porous FGSMEE plate for τxz (η = 2)

150

300

26

450 Fig. 13 Effect of porosity distribution on porous FGSMEE plate for Bz (η = 2)

150

300

450 Fig.14 Effect of porosity distribution on porous FGSMEE plate for Dz (η = 2)

(a)Vu

(b)Vo

27

(c)Vx (d) Vv Fig. 15 Comparison of porous and non-porous plate for u displacement (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 16 Comparison of porous and non-porous plate for v displacement (η = 2; α = 450)

(a)Vu

(b)Vo

28

(c)Vx Fig. 17 Comparison of porous and non-porous plate for

(a)Vu

(d) Vv (η = 2; α = 450)

(b)Vo

(c)Vx (d) Vv Fig. 18 Comparison of porous and non-porous plate for ψ (η = 2; α = 450)

(a)Vu

(b)Vo

29

(c)Vx (d) Vv Fig. 19 Comparison of porous and non-porous plate for σxx (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 20 Comparison of porous and non-porous plate for σyy (η = 2; α = 450)

(a)Vu

(b)Vo

30

(c)Vx (d) Vv Fig. 21 Comparison of porous and non-porous plate for σxy (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 22 Comparison of porous and non-porous plate for τxz (η = 2; α = 450)

(a)Vu

(b)Vo

31

(c)Vx (d) Vv Fig. 23 Comparison of porous and non-porous plate for Bz (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig.24 Comparison of porous and non-porous plate for Dz (η = 2; α = 450)

(a)

(b)

32

(c)

(d)

(e)

(f)

(g)

(h)

(i) Fig. 25 Effect of aspect ratio on (η = 2; Vu; α = 450) (a) u (b) v (c) (i) Bz (j) Dz

33

(j) z

(d) ψz (e) σxx (f) σyy (g) σxy (h) τxz

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

34

(i) Fig. 26 Effect of thickness ratio on (η = 2; Vu; α = 450) (a) u (b) v (c)

(j) z (d) ψz (e) σxx (f) σyy (g) σxy

(h) τxz (i) Bz (j) Dz

(a)

(b)

(c)

(d)

(e)

(f)

35

(g)

(h)

(i) (j) Fig.27 Effect of boundary condition on (η = 2; Vu; α = 450) (a) u (b) v (c) z (d) ψz (e) σxx (f) σyy (g) σxy (h) τxz (i) Bz (j) Dz

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig.28 Effect of porosity volume on u (η = 2; α = 450)

36

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 29 Effect of porosity volume on v (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx Fig. 30 Effect of porosity volume on

37

(d) Vv (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 31 Effect of porosity volume on ψ (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 32 Effect of porosity volume on σxx (η = 2; α = 450)

38

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig.33 Effect of porosity volume on σyy (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 34 Effect of porosity volume on σxy (η = 2; α = 450)

39

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 35 Effect of porosity volume on τxz (η = 2; α = 450)

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 36 Effect of porosity volume on Bz (η = 2; α = 450)

40

(a)Vu

(b)Vo

(c)Vx (d) Vv Fig. 37 Effect of porosity volume on Dz (η = 2; α = 450)

41

Table 1. Material properties of BaTiO3 and CoFe2O4 [35]. Material properties C11 = C22 (109 N/m2) C12 (109 N/m2) C13 = C23 (109 N/m2) C33 (109 N/m2) C44 = C55 (109 N/m2) C66 (109 N/m2) ρ (kg/m3) e31 = e32 (C/m2) e33 (C/m2) e24 = e15 (C/m2) ξ11 = ξ22 (10-9 C/Nm2) 11 = 22 (10-6 Ns2/C2) 33 (10-6 Ns2/C2) q31 = q32 (N/Am) q33 (N/Am) q24 = q15 (N/Am)

BaTiO3 166 77 78 162 43 44.5 5300 - 4.4 18.6 11.6 11.2 5 10 -

CoFe2O4 286 173 170.5 269.5 45.3 56.5 5800 0.08 - 590 157 180.3 699.7 550

Table 2. Convergence and validation studies of normalized natural frequencies of FG-MEE plate h/a

0.2

0.1

Present (44) Present (88) Present (1212) Present (1616) Present (2020) Milazzo [30] Present (44) Present (88) Present (1212) Present (1616) Present (2020) Milazzo [30]

1 6.798 6.638 6.623 6.619 6.618 6.735 9.720 9.663 9.652 9.639 9.637 9.584

2 7.923 7.894 7.872 7.863 7.860 8.223 13.598 13.421 13.417 13.413 13.408 12.852

3 11.283 11.213 11.208 11.203 11.198 11.882 14.909 14.821 14.811 14.807 14.801 14.733

Modes 4 13.595 13.487 13.462 13.458 13.455 13.463 23.158 22.986 22.963 22.959 22.952 22.577

5 13.642 13.608 13.592 13.588 13.583 15.049 27.195 26.943 26.932 26.928 26.921 25.701

6 15.202 15.183 15.172 15.167 15.161 16.951 27.290 27.102 27.082 27.078 27.069 28.339

7 18.982 18.888 18.854 18.849 18.843 19.027 28.106 27.994 27.979 27.971 27.967 28.734

8 18.693 18.601 18.592 18.583 18.579 20.178 31.411 31.387 31.372 31.367 31.361 32.391

9 19.326 19.295 19.283 19.278 19.273 20.415 35.006 34.885 34.869 34.862 34.858 36.341

Table 3: Non-dimensional frequency parameter λ  ω b2 / π 2 h (ρ/E2 )1/2 for the clamped-clamped laminated composite plate (a/h = 10). Skew Source Antisymmetric cross-ply Antisymmetric angle-ply Symmetric cross-ply angle (00/900/00/900) (450/-450/450/-450) (900/00/900/00/900) (α) Modes Modes Modes 1 2 3 1 2 3 1 2 3 Kanasogi and Ray [58] 1.4829 2.4656 3.2522 1.7974 3.3351 3.3351 1.5699 2.8917 3.7325 00 Garg et al. [59] 1.5076 2.4380 3.2254 1.8493 3.3359 3.3370 1.5635 2.4383 3.5033 Present 1.4836 2.4392 3.1328 1.8111 3.2351 3.4889 1.5314 2.4392 3.6614 Kanasogi and Ray [58] 1.5741 2.5351 3.0270 1.8313 3.2490 3.6724 1.6874 3.0458 3.9600 150 Garg et al. [59] 1.5796 2.5775 2.9892 1.8675 3.2075 3.5810 1.6571 2.9840 3.6505 Present 1.5653 2.5961 3.0730 1.8219 3.1232 3.4560 1.6261 2.8245 3.5161 Kanasogi and Ray [58] 1.8871 2.9372 3.4489 2.0270 3.4431 4.2361 2.0840 3.4023 4.6997 300 Garg et al. [59] 1.8226 2.9585 3.2357 1.9894 3.2365 4.3208 1.9596 3.1690 4.6796 Present 1.8354 3.0573 3.4428 1.9409 3.2648 4.1836 1.9262 3.1844 4.4912 Kanasogi and Ray [58] 2.5609 3.3126 4.0617 2.5609 3.3131 4.2772 2.8925 4.1906 5.4149 450 Garg et al. [59] 2.2996 3.4773 4.4889 2.3194 3.4870 4.5009 2.4811 4.4875 5.3289 Present 2.3263 3.4126 4.3756 2.2676 3.4056 4.3425 2.4260 4.1908 5.2113

42

Table 4: Non-dimensional frequency parameter λ  ω b2 / π 2 h (ρ/E 2 )1/2 for the laminated composite plate (a/h = 10). Skew Source Antisymmetric cross-ply Antisymmetric angle-ply angle (00/900/00/900) (450/-450/450/-450) (α) Modes Modes 1 2 3 1 2 3 Kanasogi and Ray [58] 1.4829 2.4656 3.2522 1.7974 3.3351 3.3351 00 Garg et al. [59] 1.5076 2.4380 3.2254 1.8493 3.3359 3.3370 Present 1.4836 2.4392 3.1328 1.8111 3.2351 3.4889 Kanasogi and Ray [58] 1.5741 2.5351 3.0270 1.8313 3.2490 3.6724 150 Garg et al. [59] 1.5796 2.5775 2.9892 1.8675 3.2075 3.5810 Present 1.5653 2.5961 3.0730 1.8219 3.1232 3.4560 Kanasogi and Ray [58] 1.8871 2.9372 3.4489 2.0270 3.4431 4.2361 300 Garg et al. [59] 1.8226 2.9585 3.2357 1.9894 3.2365 4.3208 Present 1.8354 3.0573 3.4428 1.9409 3.2648 4.1836 Kanasogi and Ray [58] 2.5609 3.3126 4.0617 2.5609 3.3131 4.2772 450 Garg et al. [59] 2.2996 3.4773 4.4889 2.3194 3.4870 4.5009 Present 2.3263 3.4126 4.3756 2.2676 3.4056 4.3425

simply supported Symmetric cross-ply (900/00/900/00/900) Modes 1 2 3 1.5699 2.8917 3.7325 1.5635 2.4383 3.5033 1.5314 2.4392 3.6614 1.6874 3.0458 3.9600 1.6571 2.9840 3.6505 1.6261 2.8245 3.5161 2.0840 3.4023 4.6997 1.9596 3.1690 4.6796 1.9262 3.1844 4.4912 2.8925 4.1906 5.4149 2.4811 4.4875 5.3289 2.4260 4.1908 5.2113

Table 5: Effect of skew angle m= 0.1, a/h=100, b/a = 1) Skew Porosity angle (α) type 1 Pure FG 6.366 Vu 6.060 150 Vo 6.272 Vx 6.157 Vv 6.067 Pure FG 8.461 Vu 8.057 300 Vo 8.345 Vx 8.177 Vv 8.068 Pure FG 12.617 Vu 12.018 450 Vo 12.454 Vx 12.187 Vv 12.034

on natural frequency of porous FGSMEE plate (SSSS, η = 0.2, porosity factor,

Table 6: Effect of skew angle m= 0.1, a/h=100, b/a = 1) Skew Porosity angle (α) type 1 Pure FG 6.235 Vu 5.922 150 Vo 6.139 Vx 6.021 Vv 5.938 Pure FG 8.265 Vu 7.850 300 Vo 8.147 Vx 7.973 Vv 7.876 Pure FG 12.309 Vu 11.691 450 Vo 11.731 Vx 11.865 Vv 11.731

on natural frequency of porous FGSMEE plate (SSSS, η = 0.5, porosity factor,

2 12.115 11.535 11.956 11.699 11.549 15.419 14.679 15.205 14.900 14.697 22.052 20.993 21.734 21.321 21.018

2 11.851 11.255 11.688 11.424 11.289 15.093 14.334 14.874 14.560 14.376 21.597 20.511 20.569 20.847 20.569

3 14.078 13.403 13.890 13.597 13.420 20.060 19.102 19.799 19.372 19.126 32.663 31.109 32.251 31.536 31.152

3 13.771 13.079 13.579 13.279 13.119 19.604 18.620 19.337 18.897 18.679 31.877 30.279 30.380 30.718 30.380

4 23.452 22.321 23.095 22.689 22.348 28.061 26.715 27.656 27.133 26.749 40.350 38.414 39.761 39.021 38.462

4 22.993 21.837 22.631 22.211 21.896 27.468 26.088 27.055 26.515 26.164 39.495 37.511 37.621 38.129 37.621

43

Modes 5 27.746 26.416 27.358 26.817 26.449 36.454 34.708 35.944 35.236 34.753 54.888 52.260 54.116 53.058 52.327

Modes 5 27.153 25.789 26.756 26.200 25.865 35.657 33.866 35.136 34.406 33.969 53.676 50.979 51.136 51.794 51.136

6 30.811 29.327 30.368 29.784 29.361 42.414 40.388 41.846 40.975 40.441 66.789 63.609 65.929 64.498 63.696

6 30.191 28.672 29.741 29.138 28.752 41.449 39.368 40.869 39.968 39.493 65.187 61.916 62.123 62.828 62.123

7 43.270 41.192 42.622 41.860 41.244 49.755 47.374 49.012 48.139 47.437 69.401 66.093 68.381 67.143 66.188

7 42.369 40.240 41.711 40.918 40.356 48.663 46.219 47.908 46.999 46.361 67.784 64.383 64.595 65.454 64.595

8 46.279 44.065 45.627 44.738 44.123 56.024 53.350 55.242 54.158 53.422 81.581 77.705 80.485 78.838 77.816

8 45.261 42.989 44.597 43.675 43.121 54.748 52.001 53.948 52.830 52.166 79.597 75.606 75.863 76.772 75.863

9 53.048 50.510 52.268 51.317 50.577 67.324 64.105 66.350 65.111 64.190 100.638 95.835 99.212 97.308 95.966

9 51.889 49.284 51.090 50.111 49.432 65.830 62.525 64.834 63.555 62.717 98.324 93.390 93.687 94.898 93.687

Table 7: Effect of skew angle on natural frequency of porous FGSMEE plate (SSSS, η = 2, porosity factor, m= 0.1, a/h=100, b/a = 1 Skew Porosity Modes angle (α) type 1 2 3 4 5 6 7 8 9 Pure FG 6.034 11.451 13.307 22.289 26.253 29.237 40.984 43.711 50.134 Vu 5.706 10.829 12.583 21.084 24.827 27.652 38.760 41.332 47.409 150 Vo 5.935 11.282 13.106 21.915 25.839 28.772 40.306 43.023 49.302 Vx 5.809 11.006 12.792 21.473 25.257 28.136 39.462 42.048 48.275 Vv 5.735 10.887 12.651 21.184 24.956 27.792 38.958 41.557 47.658 Pure FG 7.969 14.598 18.916 26.565 34.443 39.978 46.997 52.820 63.566 Vu 7.533 13.806 17.884 25.124 32.568 37.794 44.439 49.941 60.108 300 Vo 7.845 14.372 18.637 26.137 33.902 39.375 46.216 51.987 62.532 Vx 7.662 14.042 18.175 25.568 33.131 38.422 45.249 50.809 61.183 Vv 7.577 13.877 17.985 25.253 32.745 38.012 44.678 50.220 60.430 Pure FG 11.843 20.904 30.694 38.190 51.825 62.762 65.323 76.616 94.822 Vu 11.193 19.772 29.014 36.117 49.000 59.323 61.754 72.422 89.648 450 Vo 11.667 20.568 30.251 37.572 51.009 61.842 64.249 75.439 93.306 Vx 11.377 20.121 29.477 36.758 49.849 60.282 62.868 73.648 91.225 Vv 11.261 19.870 29.188 36.304 49.271 59.682 62.111 72.854 90.152 Table 8: Effect of skew angle on natural frequency of porous FGSMEE plate (SSSS, η = 5, porosity factor, m= 0.1, a/h=100, b/a = 1) Skew Porosity Modes angle (α) type 1 2 3 4 5 6 7 8 9 Pure FG 5.944 11.275 13.101 21.975 25.855 28.813 40.369 43.027 49.363 Vu 5.613 10.644 12.368 20.756 24.411 27.208 38.117 40.617 46.604 150 Vo 5.845 11.104 12.899 21.599 25.438 28.345 39.689 42.335 48.524 Vx 5.716 10.822 12.579 21.147 24.845 27.695 38.822 41.337 47.478 Vv 5.635 10.687 12.419 20.831 24.507 27.313 38.266 40.785 46.790 Pure FG 7.838 14.378 18.612 26.165 33.908 39.331 46.264 51.975 62.572 Vu 7.397 13.576 17.567 24.705 32.008 37.117 43.674 49.058 59.068 300 Vo 7.714 14.150 18.331 25.735 33.363 38.724 45.480 51.136 61.529 Vx 7.526 13.813 17.860 25.153 32.575 37.749 44.488 49.932 60.152 Vv 7.429 13.628 17.641 24.802 32.140 37.280 43.853 49.266 59.309 Pure FG 11.639 20.597 30.176 37.612 51.010 61.702 64.249 75.318 93.288 Vu 10.981 19.450 28.472 35.511 48.148 58.215 60.631 71.066 88.045 450 Vo 11.462 20.258 29.729 36.990 50.189 60.775 63.170 74.131 91.760 Vx 11.165 19.802 28.940 36.157 49.002 59.182 61.752 72.303 89.634 Vv 11.031 19.523 28.601 35.651 48.350 58.483 60.898 71.386 88.420

44

Table 9: Effect of thickness ratio on natural frequency of porous FGSMEE plate (SSSS, α = 450, η = 2, porosity factor, m= 0.1, b/a = 1) a/h Modes 1 2 3 4 5 6 7 8 9 10 8.301 13.734 17.162 19.834 20.307 20.818 23.493 28.413 30.156 20 9.641 16.141 25.487 25.522 34.326 38.402 38.966 39.672 46.989 Pure FG 50 10.994 18.488 29.038 32.362 45.733 54.245 58.041 69.906 83.114 100 11.843 20.904 30.694 38.190 51.825 62.762 65.323 76.616 94.822 10 7.853 12.995 16.242 18.761 19.219 19.702 22.230 26.901 28.546 Vu 20 9.117 15.265 24.105 24.143 32.486 36.324 36.877 37.528 44.463 50 10.392 17.483 27.451 30.610 43.246 51.307 54.876 66.079 78.603 100 11.193 19.772 29.014 36.117 49.000 59.323 61.754 72.422 89.648 10 8.161 13.497 16.709 19.307 19.930 20.436 22.871 27.843 29.571 Vo 20 9.490 15.889 25.078 25.093 33.419 37.770 38.251 38.619 45.746 50 10.829 18.206 28.612 31.825 45.018 53.303 57.158 68.840 81.613 100 11.667 20.568 30.251 37.572 51.009 61.842 64.249 75.439 93.306 10 7.997 13.240 16.708 19.303 19.607 20.096 22.869 27.487 29.147 Vx 20 9.273 15.527 24.528 24.588 33.418 36.978 37.615 38.612 45.741 50 10.564 17.777 27.895 31.167 43.990 52.282 55.795 67.189 80.158 100 11.377 20.121 29.477 36.758 49.849 60.282 62.868 73.648 91.225 10 7.889 13.052 16.252 18.777 19.294 19.781 22.246 26.989 28.648 Vv 20 9.165 15.344 24.227 24.257 32.505 36.502 37.025 37.555 44.492 50 10.453 17.576 27.611 30.760 43.476 51.561 55.183 66.474 79.006 100 11.261 19.870 29.188 36.304 49.271 59.682 62.111 72.854 90.152

45

Table 10: Effect of aspect ratio on natural frequency of porous FGSMEE plate (SSSS, α factor, m= 0.1, a/h=100) b/a Modes 1 2 3 4 5 6 7 0.5 37.168 45.862 68.340 97.869 115.791 116.137 164.465 1 11.843 20.904 30.694 38.190 51.825 62.762 65.323 Pure FG 1.5 7.321 13.981 20.881 27.029 32.750 42.269 44.201 2 5.908 10.313 18.594 19.189 25.384 32.361 41.105 2.5 5.275 8.359 14.660 17.697 22.131 27.899 40.007 3 4.912 7.209 11.990 17.261 20.320 25.282 38.781 4 4.524 5.962 9.045 16.882 18.561 22.172 36.482 0.5 35.133 43.366 64.646 92.529 109.570 109.815 155.573 1 11.193 19.772 29.014 36.117 49.000 59.323 61.754 1.5 6.920 13.221 19.738 25.549 30.958 39.968 41.782 Vu 2 5.585 9.751 17.576 18.134 23.997 30.609 38.854 2.5 4.987 7.903 13.855 16.728 20.919 26.389 37.815 3 4.643 6.815 11.332 16.316 19.207 23.910 36.731 4 4.276 5.635 8.548 15.958 17.545 20.965 34.570 0.5 36.605 45.136 67.191 96.447 113.621 114.339 160.879 1 11.667 20.568 30.251 37.572 51.009 61.842 64.249 1.5 7.209 13.765 20.577 26.604 32.271 41.552 43.564 Vo 2 5.817 10.155 18.332 18.901 25.009 31.781 40.531 2.5 5.197 8.233 14.443 17.452 21.810 27.400 39.453 3 4.841 7.101 11.814 17.025 20.029 24.847 37.946 4 4.461 5.875 8.915 16.653 18.299 21.818 35.632 0.5 35.719 44.120 65.838 94.010 111.684 111.814 158.532 1 11.377 20.121 29.477 36.758 49.849 60.282 62.868 1.5 7.037 13.446 20.056 25.991 31.458 40.713 42.447 Vx 2 5.679 9.914 17.850 18.434 24.388 31.210 39.454 2.5 5.069 8.035 14.082 16.984 21.254 26.904 38.394 3 4.717 6.928 11.516 16.563 19.511 24.360 37.593 4 4.341 5.726 8.684 16.198 17.819 21.333 35.444 0.5 35.341 43.601 64.954 93.055 110.022 110.415 156.310 1 11.261 19.870 29.188 36.304 49.271 59.682 62.111 1.5 6.960 13.291 19.856 25.701 31.143 40.184 42.030 Vv 2 5.617 9.804 17.682 18.249 24.138 30.758 39.088 2.5 5.016 7.948 13.941 16.829 21.045 26.516 38.044 3 4.670 6.854 11.402 16.414 19.323 24.031 36.824 4 4.302 5.669 8.602 16.054 17.650 21.079 34.628

46

= 450, η = 2, porosity

8 165.244 76.616 55.122 46.120 41.384 39.486 37.553 156.378 72.422 52.126 43.638 39.178 37.324 35.564 161.611 75.439 54.207 45.283 40.557 38.936 36.726 160.863 73.648 53.077 44.505 40.033 37.898 36.407 156.474 72.854 52.401 43.825 39.308 37.549 35.660

9 178.180 94.822 65.411 52.797 47.267 44.050 38.179 168.472 89.648 61.846 49.923 44.692 41.684 36.120 175.193 93.306 64.369 51.954 46.533 43.161 37.164 171.574 91.225 62.932 50.804 45.459 42.605 37.163 169.395 90.152 62.187 50.192 44.937 41.852 36.142