piezoelectric plates and shells

Accepted Manuscript A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells Chih-Ping Wu, Yan-Cheng Liu PII: DOI: Reference:

S0263-8223(16)30177-5 http://dx.doi.org/10.1016/j.compstruct.2016.03.031 COST 7329

To appear in:

Composite Structures

Received Date: Accepted Date:

4 December 2015 16 March 2016

Please cite this article as: Wu, C-P., Liu, Y-C., A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.03.031

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A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells Chih-Ping Wu *, Yan-Cheng Liu Department of Civil Engineering, National Cheng Kung University, Taiwan, ROC *Corresponding author. Fax: +886-6-2370804 E-mail address: [email protected]

Abstract The paper is to present an overview of various semi-analytical numerical methods for quasi-three-dimensional (3D) analyses of laminated composite and multilayered (or sandwiched) functionally graded elastic/piezoelectric materials (FGEMs/FGPMs) plates and shells with combinations of simply-supported, free and clamped edge conditions. This review introduces the development of various semi-analytical numerical methods incorporating 3D analytical approaches (i.e., the state space and asymptotic ones) with numerical techniques (i.e., the differential quadrature, meshless reproducing kernel and finite element ones), and their applications to the analyses of plates and shells made of advanced materials, such as the fiber-reinforced composite materials, FGEMs and FGPMs, and carbon nanotube-reinforced composite materials. Two micromechanical schemes (i.e., the rule of mixtures and Mori-Tanaka scheme) used to estimate the effective material properties of functionally graded structures are presented. The strong and weak formulations of the 3D piezoelectricity theory and their corresponding possible edge conditions for circular hollow cylinders are presented for the illustration purposes. A comparative study of the results obtained by using assorted semi-analytical numerical methods is undertaken.

Keywords: Review, Semi-analytical methods, Quasi-3D solutions, FGM, Plates, Shells

1

1.

Introduction A number of studies have carried out three-dimensional (3D) analyses of the mechanical/piezoelectric

responses of simply-supported, laminated composite plates/shells and multilayered functionally graded material (FGM) ones with various reinforcements, such as micro-scaled continuous graphite, grass and boron fibers, and nano-scaled discrete carbon nanotubes and graphene sheets. Among these, however, relatively few papers consider the 3D analyses of these structures with various boundary conditions compared to those that examine structures with fully simply-supported edges. Wu et al. [1] classified the 3D exact approaches for simply-supported structures available in the literature into four different categories, namely, the state space [2-5], series expansion [6-9], Pagano [10-13], and perturbation methods [14-17]. Moreover, the related two-dimensional (2D) advanced and refined theories have been surveyed in a number of review articles by Noor and Burton [18-20], Noor et al. [21], Carrera [22-25], Liew et al. [26], Swaminathan et al. [27], Thai and Kim [28], Sayyad and Ghugal [29], and Jha et al. [30]. Some pure 3D numerical methods have been presented for the stress, deformation and free vibration analyses of the aforementioned structures with assorted boundary conditions, such as the 3D differential quadrature (DQ) [31-34], 3D finite element (FE) [35-43] and 3D meshless methods [44-46], even though these methods are very time consuming. Based on the strong formulations of the 3D elasticity and piezoelectricity theories, the DQ techniques [47-51] are used to interpolate each field variable in the spatial coordinates, such that the 3D basic equations and the corresponding boundary conditions can be transformed into a set of simultaneously algebraic equations in terms of the nodal field variables. The 3D DQ solutions can be thus obtained by solving the resultant equations and using the weighted least squares, Gaussian elimination, or LU decomposition methods. As for the 3D FE and meshless methods, the sets of shape functions are constructed by using Lagrange (or Hermite) polynomials and meshless methods (i.e., reproducing kernel (RK) [52-54], moving least squares [55-57] and radial basis functions [58-60] approaches), respectively, and these are used to interpolate (or approximate) the spatial variations of each field variable. By substituting these kinematic models in the weak formulations of 3D elasticity/ piezoelectricity theories, a set of simultaneous algebraic equations can be obtained by performing the numerical integration, and then the corresponding quasi-3D FE and meshless solutions will be obtained.

2

Due to the fact that the pure 3D analytical approaches are mathematically complicated, and that the solution processes use with pure 3D numerical methods are very time consuming, a class of compromise approaches, so-called 3D semi-analytical numerical methods, has thus been developed for the quasi-3D analyses of the structures with various boundary conditions. In these, the above-mentioned analytical methods are incorporated with these numerical methods to reduce the mathematical complexity occurring in pure 3D analytical approaches, and reduce the time needed with pure 3D numerical methods. In this review paper we will focus on the development of semi-analytical numerical methods incorporating the state space and asymptotic approaches with the DQ, RK and FE methods and their applications to the 3D static, free vibration and buckling analyses of laminated composite and multilayered (or sandwiched) functionally graded elastic/piezoelectric material (FGEM/FGPM) plates and shells, such as the finite layer, meshless differential RK (DRK) collocation, element-free Galerkin (EFG), sampling surfaces (SaS), finite prism, state space DQ, state space FE, state space DRK, asymptotic DQ, asymptotic FE and asymptotic DRK methods. For illustration purposes, the strong and weak formulations of the 3D piezoelectric theory in the cylindrical shell coordinates and the appropriate possible boundary conditions are derived and presented by using Reissner’s mixed variational theorem (RMVT) [61, 62]. A comparative study of the results obtained by using different semi-analytical numerical methods is carried out, in which the accuracy and convergence of these methods are examined by comparing their solutions with the exact 3D and accurate 2D ones available in the literature.

2. Material properties Without loss of the generality, the material properties in the formulations derived later in the work can be considered as heterogeneously varying through the thickness coordinate. The structural behavior of single-layer homogeneous, laminated composite, multilayered (or sandwiched) FGEM/FGPM and functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) plates and shells can be studied by assuming appropriate material_property variations through the thickness coordinate of these. The material properties of the above-mentioned plates and shells are classified and given, as follows:

2.1. Single-layer homogeneous plates and shells

3

For a single-layered homogeneous plate or shell, the material properties, m ij (ζ ) , are assumed as homogeneous, independent of the thickness coordinate, and are given by m

ij

(ζ ) = constants,

(1)

in which ζ denotes the global thickness coordinate located at the mid-surface of a cylindrical shell or a plate.

2.2.

Multilayered homogeneous plates and shells. For a multilayered homogeneous plate or shell, the material properties are assumed to be layerwise

Heaviside functions and are given by Nl

mij (ζ ) = ∑ m(ijm) [H (ζ − ζ m −1 ) − H (ζ − ζ m )] ,

(2)

m =1

where H (ζ ) is the Heaviside function; ζ m −1 and ζ m denote the global thickness coordinate measured from the middle surface of the structure to the bottom and top surfaces of the mth-layer.

2.3.

Exponent-law varied, FG plates and shells. For an exponent-law varied, FG plate or shell, the material properties are assumed to vary exponentially

with the thickness coordinate and are given by mij (ζ ) = m ijb e

κp

[ (ζ

/ h )+ 0.5

]

,

(3)

where the superscript b denotes the bottom surface; κ p is the material-property gradient index which represents the degree of the material gradient along the thickness and can be determined by the values of the material properties at the top and bottom surfaces, i.e., κ p = ln (mijt / mijb ) ,

(4)

where the superscript t denotes the top surface. ln (ζ ) denotes the natural logarithm function of ζ , which is the inverse of the exponential function, eζ .

4

2.4.

Power-law varied, FG plates and shells. For a power-law varied FG plate or shell, the material properties are assumed to obey the power-law

distribution function dependent on the volume fractions of the constituents through the thickness coordinate, and are given by

mij (ζ ) = mijt Γ(ζ )+ mijb [1 − Γ(ζ )] ,

(5)

where Γ(ζ ) denotes the volume fraction and is defined as Γ(ζ ) = [(ζ / h ) + 0.5] k . k p is the power-law p

exponent which represents the degree of the material gradient along the thickness. When k p =0 and k p = ∞ , the present FG plate (or shell) reduces to a homogeneous piezoelectric plate (or shell) with material properties m ijt and m ijb , respectively. When k p =1, it represents that the material properties of the FG plate (or shell) vary linearly through the thickness coordinate.

2.5.

FG CNTRC plates and shells Due to the fact that carbon nanotubes (CNTs) possess outstanding physical, chemical and electrical

material properties, they have been added to the polymer matrix to form CNTRC plates and shells [63-68]. In recent years, the 3D [69-72] and 2D [73-77] advanced and refined analyses of CNTRC plates and shells have attracted considerable attentions. Three different distribution functions of CNTs varying in the thickness direction are often considered in the relevant papers, which are uniformly distributed (UD), and FG O- and X-type variations. Two different micromechanics models, the rule of mixtures [78, 79] and Mori-Tanaka scheme [80], have been used to estimate the effective material properties of the FG CNTRC structures varying through the thickness coordinate of these, and are given as follows.

2.5.1. The rule of mixtures According to the rule of mixtures, the through-thickness distributions of the effective material properties of the CNTRC layer are written in the following forms, E11 = λ1 ΓCNT (E11 )CNT + (Γm Em ) ,

(6a)

(λ2 / E22 ) = ΓCNT / (E22 )CNT + (Γm / Em ) ,

(6b) 5

(λ3 / G12 ) = ΓCNT / (G12 )CNT + (Γm / Gm ) ,

(6c)

in which (E11 )CNT , (E 22 )CNT , and (G12 )CNT denote the Young’s moduli and shear modulus of CNTs; E m and Gm stand for the corresponding properties of the polymer matrix; λi (i = 1 − 3) are the CNT efficiency parameters, which can be determined by equalizing the material properties of the CNTRC core obtained using the rule of mixtures and molecular dynamics simulation, and ΓCNT and Γm are the volume fractions of CNTs and polymer matrix, respectively, in which ΓCNT + Γm = 1 . The through-thickness distributions of the volume fraction of CNTs, ΓCNT , for the above-mentioned three types of CNTRC core are given as follows: * ΓCNT = ΓCNT ,

(UD-type CNTRC core),

(7a)

* ΓCNT (ζ ) = 2 1 − (2 ζ / h ) ΓCNT

]

(FG O-type CNTRC core),

(7b)

* ΓCNT (ζ ) = 2 (2 ζ / h ) ΓCNT

(FG X-type CNTRC core),

(7c)

[

* in which h denotes the thickness of the CNTRC structure, and ΓCNT denotes the volume fraction index,

and is expressed as * ΓCNT = WCNT /[WCNT + (ρCNT / ρ m ) − (ρCNT / ρ m )WCNT ] ,

(8)

in which WCNT denotes the mass fraction of CNTs in the CNTRC structure, and ρ CNT and ρ m are the mass densities of the CNT reinforcements and the polymer matrix, respectively. The Poisson’s ratio ν 12 of the FG CNTRC core is determined in the same way, as follows: * (ν 12 )CNT + (Γm ν m ) , ν 12 = ΓCNT

(9)

in which (ν 12 )CNT and ν m are the Poisson’s ratios of the CNT reinforcements and the polymer matrix, respectively. ν 12 is considered as a constant through the thickness coordinate of the FG CNTRC core, and ν 12 = ν 13 = ν 23 .

Using Eqs. (6a-e), (7a-c), (8) and (9), we can obtain the through-thickness distributions of the effective properties of the CNTRC structure.

6

2.5.2. Mori-Tanaka scheme Within the frame work of Eshelby’s elastic inclusion theory [78, 79] restricted to one single inclusion in a semi-infinite elastic, homogeneous and isotropic medium, Mori and Tanaka [80] extended the original theory to the more general cases of multiple inclusions embedded into a finite elastic medium, in which an equivalent inclusion-average stress method is used. The Mori-Tanaka micromechanics scheme was further reformulated by Benveniste [81] for the computation of the effective material properties of composites (C), and this is expressed in the following form C = Cm + ΓCNT (CCNT − Cm )A [Γm I + ΓCNT A ] , −1

(10)

in which Cm and C CNT denote the generalized stiffness tensors of the matrix and CNTs, respectively. I is the fourth-order unit tensor. A is the dilute mechanical strain concentration tensor, and is given by

[

]

A = I + S C −m1 (C CNT − C m )

−1

,

(11)

in which S stands for the fourth-order Eshelby tensor, and the nonzero components of S [82] are given as S 1111 = S 2222 = (5 c11 + c12 ) / 8 c11 ,

S 1212 = S 2121 = S 1221 = S 2112 = (3 c11 − c12 ) / 8 c11 ,

S 1313 = S 3131 = S 1331 = S 3113 = S 2323 = S 3232 = S 2332 = S 3223 = 1 / 4 , S 1122 = S 2211 = (3 c12 − c11 ) / 8 c11 ,

S 1133 = S 2233 = c13 / 2 c11 .

(12a-g)

For a two-phase isotropic material, the effective material properties are further reduced to be a set of explicit formula [83, 84], as follows:

K (ζ ) =

G (ζ ) =

Γr ( K r − K m )  Kr − K m  1 + (1 − Γr )  K m + ( 4 / 3) Gm  

Γr (Gr − Gm )  Gr − Gm  1 + (1 − Γr )  Gr + f m  

in which f m = Gm

+ Km ,

(13a)

+ Gm ,

(13b)

9 K m + 8Gm ; the subscripts m and r denote the matrix and reinforcement materials, Γr is 6 ( K m + 2Gm )

the volume fraction of the reinforcement, and K and G stand for the volume and shear moduli, respectively. 7

3.

Basic equations of 3D piezoelectricity We consider a laminated homogeneous piezoelectric or multilayered FGPM circular hollow cylinder

subjected to the electro-mechanical loads on the lateral surfaces, as shown in Fig. 1(a). A global cylindrical coordinate system ( x, θ and r coordinates) is adopted and located at the center of the cylinder, and a set of local thickness coordinates, zm (m = 1, 2, L , Nl ) , is located at the middle surface of each individual layer, as shown in Fig. 1(b), where Nl denotes the total number of layers constituting the cylinder and taken to be N l =3 as a special case. The thicknesses of each individual layer and the cylinder are hm (m = 1, 2,L, N l ) and Nl

h, while h = ∑ hm . R and L are the mid-surface radius and length of the cylinder, respectively, and m =1

r = R + ζ . The relationship between the global and local thickness coordinates in the mth-layer is ζ = ζ m + zm ,

in which ζ m = (ζ m + ζ m −1 ) / 2 , and ζ m and ζ m −1 are the global thickness coordinates measured from the middle surface of the cylinder to the top and bottom surfaces of the mth-layer, respectively. For a typical mth-layer, the linear constitutive equations, which are valid for the orthotropic piezoelectric materials, are given by ( m) σ (xm)  c11  ( m)   ( m) σ θ  c12 ( m) σ (rm)  c13  ( m)  =  τ θ r   0 τ x( mr)   0  ( m)   τ xθ   0

 D (xm )   0  (m)   D θ  =  0  D ( m )  e ( m )  r   31

( m) c12

( m) c13

0

0

( m) 22 ( m) 23

( m) 23 ( m) 33

0

0

0

0

0

0

( m) c44

0

0 0

0 0

0 0

( m) c55 0

c c

e

c c

0 0

0 0

(m) 32

(m) 33

e

0 e

(m) 24

0

0  ε (xm )   0     0  ε (θm )   0 0  ε (rm )   0   − 0  γ θ( mr)   0 ( m) 0  γ x( mr)  e15   ( m)   ( m) c66  γ xθ   0

e15(m ) 0 0

0  0 0 

0 0 0 (m ) e24

0 0

( m)  e31 ( m)  e32  ( m ) E x  ( m)  e33     E (θm )  , 0   (m)  Er  0   0 

ε (xm )   (m)  ε θ  η ( m ) 0 0  ε (rm )   11  (m) 0   ( m )  +  0 η 22 γ (m)   θr   0 0 η 33  γ ( m )   xr  (m)  γ xθ 

 E (xm )   (m)  E θ  ,  E (rm )   

where σ (xm ) , σ (θm ) , L , and τ x(mθ ) are the stress components; ε (xm) , ε (θm) , L , and γ x(θm)

(14)

(15)

are the strain

components; D (xm) , D (θm ) and D (rm ) are the electric displacement components; E (xm) , E (θm) and E (rm) are the electric field components; c ij(m) , e kl(m ) and η ll(m) are the elastic, piezoelectric and dielectric permeability

8

coefficients, respectively, which are variable through the thickness coordinate in the FGPM layer, and are constants in the homogeneous one. The strain-displacement relations for each individual layer are written as follows: ε ( mx )   ∂ x  (m )   ε θ   0 ε (rm )   0  ( m)  =  γ θ r   0 γ (xmr)   ∂ r  ( m)   −1 γ x θ   r ∂θ

0

0



(r )  u

(r )∂ −1

−1

θ

(m )    ( mx )   u  , r −1 ∂θ   (θm )  ur  ∂x    0 

∂r

0

(16)

(− r ) + ∂ ( ) −1

r

0

( )

∂x

where ∂ i = ∂ / ∂i , i = x , θ and r . The electric field-electric potential relations for each individual layer are given by  E ( mx)   − ∂ x   ( m)   −1  (m )  E θ  =  − r ∂θ  Φ ,  E ( m)   − ∂  r   r  

(

)

(17)

where Φ (m) denotes the electric potential of the mth-layer.

4.

The Reissner mixed variational theorem The Reissner mixed variational theorem is used to derive the Euler-Lagrange equations of the

multilayered FGPM cylinder and its possible boundary conditions, and its corresponding energy functional is written in the form of Nl

ΠR = ∑

∫

m =1

hm / 2 − hm / 2

∫∫ [σ Ω

(m) x

ε (mx ) + σ (θm ) ε (θm ) + σ ( mr ) ε ( mr ) + τ (xrm ) γ (xrm ) + τ (θmr) γ (θmr) + τ (xmθ) γ (xmθ)

)]

(

− D ( mx ) E ( mx ) − D (θm ) E (θm ) − D ( rm ) E ( mr ) − B σ (ijm ) , D ( km ) r dx d θ dzm − ∫∫

q ( x, θ ) u

− ∫∫

Dr Φ − Φ

± r

Ω σ±

(

Ω φ±

Nl

−∑ m =1 Nl

−∑ m =1 Nl

−∑ m =1

∫

hm / 2 − hm / 2 hm / 2

∫

− hm / 2

∫

− hm / 2

hm / 2

±

∫ (t Γσ

± r

(R ± h / 2 ) dx dθ − ∫∫Ω

∫

Γφ

± d

±

(R ± h / 2) dx dθ

) (R ± h / 2 ) dx dθ

(m) x

∫ [(u Γu

D ( x, θ ) Φ ± r

(18)

)

u ( mx ) + t (θm ) u (θm ) + t (rm ) u ( mr ) dΓ dz m − ∫

(m) x

(

hm / 2 − hm / 2

∫ (D Γd

(m) n

]

Φ ( m ) ) d Γ dzm

− u (xm ) ) t ( mx ) + (u (θm ) − u (θm ) ) t (θm ) + (u ( mr ) − u (rm ) ) t ( mr ) dΓ dz m

D (nm ) Φ ( m ) − Φ

(m)

) dΓ dz

m

,

9

where Ω denotes the cylinder domain on the x − θ surface, and Ω ± denotes the outer surface ( ζ = h / 2 ) and the inner one ( ζ = − h / 2 ) of the cylinder, in which the transverse load ( q r± ), and electric potential ( Φ ± ) or normal electric displacement ( D r± ) are applied; Γ σ , Γ u , Γ φ

and Γ d denote the portions of the edge

boundary, where the surface traction, elastic displacement, electric potential and normal electric displacement components are prescribed, respectively (i.e., t i = t i and u i = u i , in which i= x, θ and r; Φ = Φ ; Dn = Dn ); B (σ ij , D k ) is the complementary energy density function, in which ε (mx ) = ∂B / ∂σ (mx ) , ε (θm ) = ∂B / ∂σ (θm ) , ε (mr) = ∂ B / ∂σ (mr) , γ (mxr) = ∂ B / ∂τ (xrm) , γ (θm)r = ∂ B / ∂τ (θmr) ,

γ (xmθ) = ∂ B / ∂τ (mxθ) , E(mx) = −∂B / ∂D(mx) ,

E (θm ) = −∂B / ∂D (θm) , and E ( mr) = −∂B / ∂D ( mr) .

In the RMVT-based formulation, we take the elastic displacement, transverse normal and shear stress, normal electric displacement and electric potential components as the primary variables subject to variation.

4.1. The RMVT-based strong formulation

Substituting Eqs. (14)_(17) into Eq. (18), imposing the stationary principle of the Reissner energy functional (i.e., δ Π R = 0 ), and performing the integration by parts with using Green’s theorem, we can finally obtain the Euler_Lagrange equations of 3D piezoelectricity from the domain integral terms and the possible boundary conditions from the boundary integral terms, which are written as follows: The Euler_Lagrange equations are ) (m ) ( m) −1 (m ) −1 ( m) δ u (m x : τ xr , r = −σ x , x −(r )τ x θ ,θ − (r )τ x r ,

(19)

) (m) (m ) −1 )σ (θm ) ,θ −(2 r −1 )τ (θmr) , δ u (m θ : τ θ r , r = −τ x θ , x − (r

(20)

( )

( )

( )

(m) (m) −1 ) δ u (m τ (θmr) ,θ − r −1 σ (rm ) + r −1 σ (θm ) , r : σ r , r = −τ x r , x − r

(21)

) ( m) ( m) ( m) ( m) (m ) ( m) ( m) δ τ (m ,x , x r : u x , r = −u r , x + (c55 ) τ xr − (e 15 / c 55 )Φ

(22)

) ( m) −1 ( m) −1 (m ) (m ) ( m) −1 ( m ) ( m) (m ) δ τ (m ,θ , θ r : u θ , r = (r )u θ − (r )u r ,θ +(c44 ) τ θ r − (r e 24 / c 44 )Φ

(23)

−1

−1

10

δσ (mr) : u ( mr) , r = − a (1m) u (xm ) , x − (a (2m) r −1 )u (θm) ,θ −(a (2m )r −1 )u ( mr) + η 33(m ) σ ( mr) + e (33m ) D (rm ) ,

(24)

) (m ) δ D (m , r = −b (1m )u (xm) , x −(b (2m ) r −1 )u (θm) ,θ −(b (2m ) r −1 )u (rm) + e (33m ) σ (rm) − c (33m ) D (rm ) , r : Φ

(25)

( )

( )

δ Φ (m) : D (rm) ,r = − D (xm ) , x − r −1 D (θm ) ,θ − r −1 D (rm ) ,

(26)

in which m = 1, 2, L , N l ; and

(

)(

)

(i=1 and 2),

(

)(

)

(i=1 and 2),

a ( mi ) = c (im3 ) η (33m ) + e(3mi) e (33m ) / c (33m) η (33m ) + e(33m) e (33m ) b ( mi ) = c (im3 ) e(33m) − e(3mi) c (33m) / c (33m ) η (33m) + e (33m ) e(33m)

( m) η 33 = η (33m ) / (c (33m ) η (33m) + e (33m ) e(33m) ) , e (33m ) = e (33m ) / (c (33m ) η (33m) + e (33m ) e(33m) ), and c (33m ) = c (33m) / (c (33m) η (33m ) + e(33m ) e(33m) ) .

(27a-e) The lateral boundary conditions are obtained, as follows: For the open-circuit surface conditions,

[τ

( Nl ) xr

τ (θNr )

σ (rN )

[τ

(1) xr

τ (θ1)r

σ (r1)

l

l

] [

D( rN l ) = 0

] [

D (r1) = 0

q +r

D +r

]

q −r

D r−

]

0

0

on z N = h N / 2 (or ζ = h / 2 ), l

l

(28a)

on z1 = − h1 / 2 (or ζ = − h / 2 );

(28b)

on z N = h N / 2 (or ζ = h / 2 ),

(29a)

on z1 = − h1 / 2 (or ζ = − h / 2 );

(29b)

For the closed-circuit surface conditions,

[τ

( Nl ) xr

[τ

(1) xr

τ (θNr ) l

τ (θ1)r

σ (rN ) l

σ (r1)

] [

Φ( Nl ) = 0

] [

Φ (1) = 0

qr+

0

0

q −r

Φ+

Φ

−

]

]

l

l

The possible edge boundary conditions are obtained, as follows:

σ x(m ) n1 + τ (xmθ) n2 = t x(m )

or

u x( m ) = u x( m ) ,

(30a)

τ (xmθ) n1 + σ (θm ) n2 = t θ( m )

or

u (θm) = u θ( m) ,

(30b)

τ (xmr) n1 + τ (θmr) n2 = t (rm)

or

u (rm) = u (rm) ,

(30c)

D (xm ) n1 + D (θm ) n2 = D (nm)

or

Φ (m ) = Φ (m ) ,

(30d)

11

where m = 1, 2, L , N l , and n1 and n 2 stand for components of the unit normal vectors on the edges. These sets of Euler_Lagrange Eqs. (19)-(26) associated with a set of appropriate boundary conditions (Eqs. (30a)-(30d)) are composed of a well-posed boundary value problem, which is the so-called strong formulation of this problem. Introducing the numerical models of various primary variables, we may develop the corresponding strong formulation-based semi-analytical numerical methods, such as meshless DRK collocation, SaS, state space DQ, state space DRK, asymptotic DQ and asymptotic DRK methods. After the primary variables are determined, the other secondary variables can be obtained using the determined primary variables, as follows: σ ( mp ) = Q ( mp )L 1 u ( m ) + Q (wm ) L 2 u ( mr ) + Q (am ) σ ( mr ) + Q (bm ) D (rm ) ,

(31)

D( mp) = S(em )σ (sm ) + S ( mη ) L3 Φ ( m ) ,

(32)

in which σ (pm) = {σ x( m)

Q (mp )

σ θ( m ) τ x( mθ )

Q (11m ) Q (12m)  = Q (21m ) Q (22m)  0 0 

(

 e( m ) / c ( m ) S(em ) =  15 55 0 

 ∂x L1 =  0  r −1 ∂θ

( )

)

}, T

 D (m )  D( mp) =  (xm )  , u (m) = u x(m)  D θ 

Q (12m ) 0  0   (m ) 0 , Qw =  0 Q (22m) (m )   0 Q 66  0 

0  0 , 0 

(

)

( ) ( ) ,

 r −1   r −1 ∂θ  , L 2 =  r −1  0 ∂ x   0

( )

Q (ijm) = c (ijm) − a ( mj ) c (im3 ) − b (jm ) e(3mi)

 

}

Q (am )

  η (11m) + e (15m ) e(15m) / c (55m) 0 ( m) , S =   η e(24m) / c (44m )  0 

(

τ (xmr)  T uθ( m) , σ (sm) =  ( m )  , τ θ r 

{

a (1m)  b (1m)   ( m)    = a 2  , Q (bm ) = b ( m2 )  ,  0   0     

)

0

(η

( m) 22

+e

( m) 24

(m ) 24

e

/c

(m) 44

 , 

)

 −∂  L 3 =  −1 x  ,  − r ∂θ 

( )

m) Q ( 66 = c ( m66) ,

(i, j=1 and 2),

c~i(3m ) = c (im3 ) / c (33m)

(i=1 and 2).

4.2. The RMVT-based weak formulation

As we mentioned above, the Reissner energy functional of the 3D piezoelectricity problem (i.e., Π R ) is written in Eq. (18). In order to derive its corresponding weak formulation, we take the first-order variation of this functional to be zero and let the differentiation be equally distributed among the primary field variables and their variations, which leads to

12

Nl

δ ΠR = ∑∫

∫∫ {(δε ) σ

hm / 2

− hm / 2

m =1

(m ) T p

Ω

[ [E

(

(m ) p

+ δε s( m )

)σ T

∫∫ − ∫∫ −

Ω σ±

q

Ω φ±

∑∫ m =1

∑∫ m =1

hm / 2

hm / 2 −h m / 2

Nl

−

∑∫ m =1

±

− hm / 2

Nl

−

+ e (33m )

(R ± h / 2 ) δ u

± r

(Φ

Nl

−

(m ) r

hm / 2 − hm / 2

± r

σ (rm ) + (δσ s( m ) ) (ε s( m ) − S σ( m )σ s( m ) − S (em ) E (sm ) ) T

(m ) r

+ δε

( ) ε −e D − (Q ) ε −c D σ dx d θ − ∫∫ D (R ± h / 2 ) δ Φ Τ

( m) p

( m) 33

(m ) r

(m) Τ b

(m ) p

( m) 33

(m ) r

(m) + δσ r( m ) ε (rm ) − η 33 σ (rm ) + Q (am )

− δD (rm )

(m ) s

(m) r

± r

Ω ±d

±

]− (δ E ) D ]} r dx dθ dz (m ) T s

(m) s

− δE (rm ) D (rm )

m

dx dθ

− Φ ± ) (R ± h / 2 ) δ D r± dx d θ

∫ (t Γσ

(33)

δ u (xm ) + t (θm ) δ u (θm ) + t (rm ) δ u ( mr ) ) d Γ dz m

(m ) x

∫ [(u

(m) x

− u (xm ) ) δ t (xm ) + (u (θm ) − u (θm ) ) δ t (θm ) + (u ( mr ) − u (rm ) ) δ t (rm ) dΓ dz m

∫ (Φ

(m)

−Φ

Γu

]

Γφ

(m)

) δD

Nl

(m ) n

∑∫

dΓ dz m −

m =1

hm / 2 − hm / 2

∫ (D

δ Φ (µ) ) d Γ dz m ,

(m) n

Γd

where the superscript of T denotes the transposition of the matrices or vectors; and Γu , Γφ , and Γσ , Γd stand for the boundary edges, in which the essential and natural conditions are prescribed. In addition,

[

ε (pm) = ε x(m ) ε (θm)

[

ε s( m) = γ x( mr)

γ (θmr)

γ x(mθ )

]

T

]

[

[

τ (θmr)

]

T

Ε (rm) = −B (9m) Φ ( m ) ,

= B 1( m ) u ( m ) + B (2m ) w ( m) ,

= B (3m ) u ( m ) + B (4m) w ( m) ,

σ (pm) = σ (xm ) σ (θm ) τ (xmθ)

σ s( m) = τ x( mr)

T

]

T

= Q (pm) B 1( m) u ( m ) + Q (pm) B (2m) w ( m ) + Q (am) B (7m ) σ ( m ) + Q (bm) B (10m) D ( m ) ,

= B (6m ) τ ( m) ,

[

D (sm ) = D (xm)

D (θm )

[

[

]

i =1, 2,L, nφ

(

B

( m) 1

)

, +1

 ψ u( m) i ∂ x  = 0 (1 / r ) ψ ( m) ∂ u i θ 

(

(

 Dψ u(m ) B (3m) =  0 

)

)

i

[(

]

]

T

i =1, 2 ,L, n w

D (m ) = D3( m)

)]

i i =1, 2 ,L, n d

, +1

)

(m ) u i

)

0

(− 1 / r ) (ψ

) + (Dψ )

(m ) u i

E (θm )

( m) u i

]

T

= −B (8m ) Φ ( m ) ,

D (rm) = B (10m) D ( m) ,

( ) ( )

 τ 13( m) i  , σ ( m) = σ 3( m) ( m)   τ 23 i  i =1, 2 ,L, nτ +1

, τ ( m) =  +1

  (1 / r ) ψ ∂θ  ψ u(m ) i ∂ x  i =1, 2 ,L, n

(

Ε (sm ) = E (xm)

= S (em ) B (6m )τ ( m) − S η( m ) B (8m) Φ ( m ) ,

0

(

[

σ (rm ) = B (7m ) σ ( m ) ,

u ( m )  u ( m) =  i( m )  , w (m ) = wi(m ) v  i  i =1, 2,L,nu +1

Φ ( m) = Φ i(m )

ε (rm ) = B (5m) w (m ) ,

, u

+1

   i =1, 2 ,L, n

13

(

[(

( m)  1 / c55 Sσ(m) =   0

)

0  , ( m)  1 / c44 

(

0  B (2m) = (1 / r ) ψ (wm ) 0 

(

, u +1

)



)  i

 i =1, 2,L, n

, w +1

)]

i i =1, 2 ,L, nσ +1

,

(

)

 ψ w( m) i ∂ x  B (4m) =   ( m) (1 / r ) ψ w i ∂ θ  i =1, 2 ,L, n

(

[(

)]

[(

)]

B (7m) = ψ (σm )

B (10m) = ψ (dm )

)

i i =1, 2 ,L, nσ

[(

B (5m) = Dψ w( m)

,

)]

i i =1, 2 ,L, n w +1

w +1

(

)

 ψ φ( m) ∂ x  i B (8m) =  ,  ( m) (1 / r ) ψ φ i ∂ θ  i =1, 2 ,L, nφ +1

, +1

(

 ψ ( m) B (6m) =  τ  0

,

(

[(

B (9m ) = D ψ φ( m)

)

)

i

0 τ

)]

 ,   i =1, 2 ,L, nτ +1 i

(ψ ) ( m)

i i =1, 2 ,L, nφ +1

,

i i =1, 2 ,L, n d +1

Imposing the stationary principle of the Reissner energy functional (i.e., δ Π R = 0 ), we obtain the corresponding system equations of the cylinder, as follows:

 K (ImI)  (m )  K II I N  (m) K  (IIIm)I ∑ m =1  K IV I  K (m )  VI  0

K (ImII)

l

K (ImIV)

K (ImV)

(m ) II III ( m) III III

(m ) II IV

(m ) II V

K

K

K

K K

hm / 2

K (IImI) =

∫

hm / 2

K (ImIV) =

∫

K (IIm)II =

∫

K (IIm)IV =

∫

K (IIIm)III =

∫

− hm / 2

(m ) T 1

0 K K

0

K

(m ) IV V (m ) V V (m ) VI V

B1(m ) r dzm , K (ImII) =

∫

K (ImV) =

( m) T 1

(m ) T 2

[(B

hm / 2

hm / 2

− hm / 2

(m ) 1

m

B (7m ) r dzm ,

( m) p

(B )

S σ( m ) B (6m) r dz m ,

hm / 2

( m) T 6

(

(m ) (B(7m) ) B(7m) r dzm , η 33 T

(

)

T

c (33m ) B(10m) B(10m ) r dzm ,

hm / 2

− hm / 2

(B ) S (m ) T 8

(m )

η

∫

∫

)

T

hm / 2

− hm / 2

∫

hm / 2

( m) T 3

( m) T 4

]

∫

K (IVm)V = −

K (Vm)VI =

∫

hm / 2

− hm / 2

14

B(10m) r dzm ,

(m ) 6

r dzm ,

hm / 2

∫

(B ) S

(m ) e

− hm / 2

(B )

(m) T 2

B(8m ) r dzm ,

( )

T

− hm / 2

hm / 2

B(8m ) r dzm ;

hm / 2

B (6m ) r dz m ,

K (IIm)V =

(m) T 6

− hm / 2

∫

( m) b

(B ) B

B (7m ) r dz m ,

l

0 0 P +  P −   σ  σ 0 0   + δ m1   , 0   0 0 0  +  − Pd  Pd 

B (2m ) r dzm ,

(B )

(m ) T 1

− hm / 2

K (IIIm)VI =

(m ) p

(B ) Q

hm / 2

− hm / 2

) K (IImIII =

Q (am) B (7m) + B (5m )

(m ) T 2

( m) T 1

K (I mIII) =

B (2m) r dz m ,

)

− hm / 2

− hm / 2

(m ) a

(B ) Q

− hm / 2

hm / 2

( m) T p

(B ) Q

hm / 2

− hm / 2

(B ) Q

− hm / 2

∫

0 (m ) IV IV ( m) V IV

,

(m ) T 2

hm / 2

K (VIm ) VI = −

( m) p

  u ( m)    0  w ( m)  K (IIIm )VI   τ (m )    = δ mN 0   σ (m )  K (Vm)VI   D(m )    K (VIm )VI  Φ (m )  0

(B ) (Q ) (B )r dz

hm / 2

∫

K

K K

K (VIm)III

(B ) Q

− hm / 2

K (IVm)IV = −

K

0 0

0

− hm / 2

∫

K (ImIII)

K

where K (ImI) = ∫

K (Vm)V =

( m) II II (m ) III II ( m) IV II ( m) V II

e (33m) B(7m) B(10m) r dzm ,

(B )

( m) T 10

B (9m ) r dz m ,

Q (bm ) B (10m ) r dz m ,

(34)

In Eq. (34), K (i mj) = (K (jmi) )

( i, j = I, II, III, IV, V, VI ), except for K (IIm)I ≠ (K (ImII) ) , because Qij(m ) ≠ Q (jim ) (i,

T

T

j =1, 2 and 6) for a piezoelectric material layer; and

[

( )

P σ+ = q m+ˆ nˆ (R + h / 2 ) δ i (n

(

)

[

w +1

P +d = D3+mˆ nˆ (R + h / 2 ) δ i (n

d

)

]

i =1, 2,L, n w +1

]

+1) i =1, 2,L, n +1 d

[ ]

( )

P σ− = q m−ˆ nˆ (R − h / 2 ) δ i 1

,

,

(

)

i =1, 2 ,L, n w +1

[ ]

P −d = D3−mˆ nˆ (R − h / 2 ) δ i 1

,

i =1, 2 ,L, n d +1

;

and the symbol of δ m l denotes the Kronecker delta functions. Equation (34) represents a set of weak formulation-based RMVT system equations, in which the continuity conditions of elastic displacement, transverse stress, electric potential and normal electric displacement components at the interfaces between adjacent layers are directly imposed in the assembly process, and thus satisfied a priori for these equations. Introducing the numerical models of primary variables to Eq. (34), we may develop the corresponding weak formulation-based semi-analytical numerical methods, such as finite layer, finite prism, EFG, state space FE and asymptotic FE methods. The primary variables induced in the cylinder can then be determined by solving Eq. (34). Subsequently, the secondary variables can be obtained using Eqs. (31) and (32).

5.

Various semi-analytical numerical methods

5.1 Fully simply-supported boundary conditions 5.1.1.

Finite layer methods

Based on the principle of virtual displacement (PVD), Cheung and Chakrabarti [85] first developed the finite rectangular layer methods (FRLMs) for the free vibration analysis of thick, layered rectangular plates, and the FRLMs were then extended to the buckling analysis of sandwich plates, static analysis of elastic thin-walled structures and static analysis of piezoelectric composite laminates by Cheung et al. [86, 87] and Cheung and Jiang [88], respectively. In the FRLMs, the simply-supported laminate is divided into a number of finite layers, and the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the field variables, respectively, for each individual layer. The 3D elasticity and piezoelectricity problems of simply-supported, multilayered rectangular plates were effectively reduced to 1D ones, in which the dependent variables in the resulting equations will be functions of thickness only. The

15

1D FE method was then used to obtain the quasi-3D solutions for the simply-supported plates. It has been demonstrated that the semi-analytical FRLMs are more effective in reducing computational effort and core requirements for simply supported laminates. These PVD-based FRLMs were also extended to the 3D static, vibration, stability and thermal buckling analyses of piezoelectric composite plates and piezoelectric antisymmetric angle-ply laminates ones by Akhras and Li [89-92]. Numerous papers have reported that the RMVT-based theoretical and numerical methods are superior to their corresponding PVD-based counterparts, such as Carrera and Brischetto [93], Brischetto and Carrera [94-96]. In the former, not only the displacement components but also the transverse stress components are taken to be the primary variables, such that the transverse stress and displacement continuity conditions at the interfaces between the adjacent layers will be satisfied a priori in their formulation; while in the latter only the displacement components are regarded as the primary variables, such that the transverse stress continuity conditions will be discontinuous across the interfaces. These PVD-based theories might give accurate predictions of the displacement and in-plane stress components of the deformed laminated composite structures and multilayered FGEM ones, while they might fail to yield the same accuracy for the transverse stresses. A post-correction process for the calculation of transverse stresses is thus usually needed to improve the accuracy for these stresses, which utilizes the indefinite integrations derived from the stress equilibrium equations. Wu and Li [97] developed a unified formulation of RMVT-based FRLMs for the static analysis of simply-supported, laminated composite plates and multilayered FGEM ones. By adopting an h-refinement to obtain the convergent results, the lower-order Lagrange polynomials, such as linear, quadratic and cubic orders, are used to interpolate the through-thickness variations of primary variables in each layer element, and the related orders used to expand the primary variables through the thickness coordinate can be freely chosen. The accuracy and convergence rate of a variety of RMVT-based and PVD-based FRLMs were assessed by comparing their solutions with the exact 3D ones available in the literature. Subsequently, the unified formulation of FRLMs was extended to the free vibration [98] and buckling [99] analyses of laminated composite and multilayered FGEM plates and their counterparts for multilayered FGPM plates and carbon nanotube-reinforced composite (CNTRC) plates with surface-bonded piezoelectric actuators and sensor layers [100-102].

16

Wu and Chang [103] formulated an RMVT-based finite cylindrical layer method (FCLM) for the static analysis of sandwich circular hollow cylinders with an embedded FGEM core by using the cylindrical coordinate system. In the implementations of these RMVT-based FCLMs, Wu and Chang found that the results obtained by letting the related orders used for expanding the primary variables through the thickness coordinate be identical to one another will be more accurate than those obtained using the orders different from one another. The convergence rates of the displacement, in-surface stress and transverse normal stress components are (2.1, 1.6, 2.0), (3.6, 1.9, 2.1) and (3.7, 3.2, 3.4) for linear, quadratic and cubic layer elements, respectively. The RMVT-based FCLMs were also extended to the various mechanical analyses of sandwich FGEM circular hollow cylinders, such as bending [104], free vibration [105] and buckling due to the applied external pressure and axial compression [106, 107].

5.1.2.

Meshless DRK collocation and EFG methods

Wu et al. [108, 109] and Yang et al. [110] proposed a DRK approximation for the 2D analysis of elastic solids and quasi-3D static analyses of multilayered piezoelectric plates and FG magneto-electro-elastic shells. The novelty of this DRK approximation is in the determination of the shape functions of derivatives of the conventional RK approximation. They are determined using a set of differential reproducing conditions without directly differentiating the RK approximation, as is necessary in the conventional RK approximation. It has been shown that the solutions of this DRK approximation-based collocation method are very close to the available 3D solutions and have a fast convergence rate. The DRK approximation is similar to other RK ones, in that the shape functions of the PK approximations at each sampling node will not possess the Kronecker delta properties. This may cause difficulties and inconvenience when the essential boundary conditions are imposed in the implementation of these methods, especially when solving the weak formulation of physical problems. Wang et al. [111] and Chen et al. [112] thus proposed the DRK interpolation and its related meshless DRK collocation and EFG methods for the 2D potential, elasticity [113], and quasi-3D elasticity problems of elastic solids and FGEM/FGPM plates [114, 115] and shells [116-119], in which the shape function of the DRK interpolation is separated into a primitive function and an enrichment function. The primitive function is required to satisfy the Kronecker delta properties by assigning a quartic spline function for which its support size at each sampling node will not cover any neighboring node, and the enrichment function is determined using a set of 17

reproducing conditions.

5.1.3. Sampling surface methods

Kulikov and Plotnikova [120] proposed a semi-analytical numerical method, the so-called sampling surfaces (SaS) method, for the quasi-3D bending analysis of simply-supported, laminated composite plates. In the SaS formulation, the plate is artificially divided into a series of Nr sampling surfaces located at the roots of an rth-order Chebyshev polynomial in the thickness coordinate. The through-thickness distributions of the field variables are interpolated as the Lagrange polynomials on the basis of the corresponding values of the set of SaS nodal surfaces. An obvious advantage of the SaS method with using the Chebyshev polynomial nodal surfaces, rather than the uniform spacing nodal surfaces, is to avoid the numerical instability that often occurs in the SaS with uniform spacing when the number of nodal surfaces sampled is greater than 21. The SaS method has been shown to converge rapidly, and its convergent results are in excellent agreement with the exact 3D solutions available in the literature. Due to the excellent performance of the SaS method, Kulikov and Plotnikova extended the SaS method to the quasi-3D coupled thermo-elastic analysis of laminated composite and FGEM structures [121-125], and the quasi-3D coupled thermos-electro-elastic analysis of laminated piezoelectric composite and FGPM structures [126-131].

5.2 . Combinations of simply-supported, free and clamped boundary conditions 5.2.1.

Finite prism methods

As mentioned above, the FRLMs and FCLMs can use to analyze the piezo-thermo-mechanical behaviors of laminated composite and multilayered FGEM/FGPM plates and shells with simply-supported edge conditions only, rather than for these structures with combinations of simply-supported, free and clamped edge conditions. Wu and Li [132, 133] thus developed the finite rectangular and cylindrical prism methods (FRPMs and FCPMs) for the analyses of multilayered (or sandwiched) FGEM plates and cylinders with various boundary conditions. In their formulations, the primary variables are expanded as the single Fourier series functions in the θ (or y) coordinate and Lagrange polynomials are used to interpolate their variations in the x − ζ domain. The FRPMs and FCPMs are different from the conventional layerwise shell (or plate) element methods, in which the power series functions and Lagrange polynomials are used to interpolate the

18

variations of primary variables in the thickness direction and in-surface element domain, respectively. Comparisons with regard to the meshes, configuration, interpolation functions, and discretized domains among the present FCPMs, layerwise shell element methods and FCLMs are tabulated in Table 1 [133].

5.2.2.

State-space meshless methods

State space methods (also called the transfer matrix method [134] and initial function method [135]) have been proposed for the exact 3D analysis of simply-supported, multilayered medium and thick plates and shells. In these, the in-plane variations of primary variables are expanded as double Fourier series functions to satisfy the boundary conditions, and the transfer matrix method is then developed to determine their variations in the thickness coordinate. In order to extend the application of the method to structures with various boundary conditions and nonhomogeneous material properties, some meshless methods are used to interpolate or approximate the variations of primary variables over the in- surface domain, such as DQ and DRK methods. Chen and Lue [136] proposed a state-space DQ method for the free vibration analysis of laminated composite plates with one pair of opposite edges being simply supported while the other pair can be combinations of simply-supported, free and clamped edges, in which the DQ technique is incorporated into the state space equations. In the state space DQ method, the state space variables are expressed in terms of the single Fourier series functions in the coordinate direction between the pair of simply-supported edges, applying the DQ technique with respect to the coordinate direction between the other two edges, and the resulting state space equations in terms of the thickness coordinate can then be exactly solved. As a result, the state space DQ method is extended to various mechanical analyses of laminated elastic/piezoelectric composite plates and shells and multilayered FGEM/FGPM ones. A partial publication list of relevant studies [137-170] is classified and tabulated in Table 1. By replacing the DQ technique with the DRK interpolation [111], Wu and Jiang [171, 172] developed the state space DRK method for the quasi-3D static analysis of sandwich FGEM and quasi-3D buckling analysis of CNTRC circular hollow cylinders with various boundary conditions. In their formulation, the DRK interpolants are used to interpolate the primary variables in the axial coordinate, and the modified Pagano method [173-177] is used to obtain the corresponding quasi-3D solutions layer-by-layer. A

19

parametric study is undertaken of the influence of the boundary conditions, mid-surface radius-to-thickness ratio, material-property gradient indices on the bending behaviors and critical load parameters of these structures. Some other meshless methods were also combined with the state space method to study the aforementioned 3D elastic problems. In conjunction with the radial point interpolation and state space methods, Li et al. [178] developed a 3D semi-analytical model for the analysis of laminated composite plates with a stepped lap repair. Introducing the reproducing kernel particle method in the 3D state space framework, Khezri et al. [179, 180] investigated the 3D static analysis of thick laminated composite plates.

5.2.3.

State space FE methods

Based on a mixed variational principle, Sheng and Ye [181-183] developed a state space FE method for the quasi-3D static analysis of laminated composite rectangular plates and cylindrical shells. In the formulation, finite element approximation is introduced for the in-plane variations of displacement and stress components, while the through-thickness distributions of these are obtained by using the state space method. The state space FE method was further applied to examine the free edge effects in the cross-ply laminated plates by Ye et al. [184]. It is shown that the through-thickness distributions of transverse stress and displacement components obtained using the state space FE solutions are continuous and that very good estimations were obtained for the stress singularities in the vicinity of free edges. The state space FE method was extended to the quasi-3D analysis of smart composites and FGM analysis by Andrianarison and Benjeddou [185].

5.2.4.

Asymptotic meshless methods

In conjunction with the DQ and perturbation methods, Wu and Lo [186] and Wu and Tsai [187] developed an asymptotic DQ method to investigate the quasi-3D static behaviors of laminated composite and FGEM annular spherical shells. After performing proper nondimensionalization, introducing a geometrical perturbation parameter, and carrying out the asymptotical expansion to the field variables, they decomposed the basic 3D equations in a series of differential equations for various orders. Upon successively integrating the resulting equations through the thickness direction, the recursive sets of governing equations for various

20

orders can be obtained. The DQ method is adopted to solve the boundary-value problems for various orders. Because the differential operators of the governing equations remain the same for various orders and the nonhomogeneous terms at the higher-order level can be calculated from the lower-order solutions, the solution procedure for the leading order problem can be repeatedly applied for the higher-order problems, which will be less time consuming. The leading-order DQ solutions can be modified order-by-order, and finally these will approach the exact 3D solutions. It has been shown that the asymptotic DQ solutions converge rapidly. The convergent solutions are obtained at the ∈2 -order level in the cases of thin shells (R/h=20), at the ∈4 -order level in the cases of moderately thick shells (R/h=10), and at the ∈6 -order level in the cases of thick shells (R/h=5). Subsequently, the asymptotic DQ method was also extended to the static, elastic buckling, thermal buckling and thermally induced dynamic instability of laminated composite conical shells by Wu et al. [188], Wu and Chen [189], Wu and Chiu [190], and Wu and Chiu [191], respectively. By replacing the DQ method with the DRK interpolation method, and incorporating the perturbation method, Wu and Jiang [192, 193] proposed the asymptotic DRK method for the quasi-3D static and free vibration analyses of sandwich FGEM cylinders with combinations of simply-supported and clamped edges. In the formulation, the through-thickness distributions of the material properties of the FGM layer are assumed to obey a power-law function according to the volume fractions of constituents composed of the layer, and the effective material properties are estimated by using the rule of mixtures and Mori-Tanaka scheme.

5.2.5.

Asymptotic FE methods

Within the framework of 3D elasticity, Tarn et al. [194], Tarn [195] and Tarn and Wang [196] developed the asymptotic FEMs for the static and dynamic analyses of multilayered anisotropic plates. By means of asymptotic expansions, they decomposed the Reissner’s functional of the plate into a series of functionals, with which the system equations of finite element models for the Kirchhoff-Love and Mindin-Reissner plate theories can be constructed as those of the leading-order problem. The stiffness matrix of the leading order problem remains unchanged from that of the higher-order problems, although with the different nonhomogeneous terms that can be obtained by using the known lower-order solutions. The quasi-3D solutions of the stress and displacement components induced by the loaded plate can thus be obtained

21

level-by-level in a hierarchical and consistent manner. Based on the aforementioned asymptotic formulation, Wu et al. [197] presented an asymptotic finite strip method (FSM) for the analysis of doubly curved laminated shells by independently interpolating the field variables as a finite series of products of trigonometric functions and crosswise polynomial functions. The first-order shear deformation theory (FSDT)-based FSM solution is shown to be the first-order approximation to the results of the asymptotic FSM, and the accuracy of the FSDT-based FSM solution can be improved level-by-level without reformulation.

6.

Illustrative Examples

A benchmark bending analysis of a simply supported, three-layered orthotropic circular hollow cylinder subjected to a sinusoidally distributed load ( q r+ = 0 and q r− = − q 0 sin (π x / L ) cos 4θ ) was presented by Varadan and Bhaskar [198], and used to validate the accuracy and convergence of various semi-analytical numerical methods. The lay-up arrangement from the inner to outer layers and geometric parameters of the cylinders is [90 0 / 0 0 / 90 0 ] , L / R = 4 and S= R / h =4, as shown in Table 3. The material properties of the cylinders are given as E L = 25x10 6 psi (172.4 GPa ) ,

E T = 1.0x10 6 psi (6.89 GPa) ,

G LT = 0.5x10 6 psi (3.45 GPa ) ,

GTT = 0.2 x10 6 psi (1.378 GPa ) ,

(35a-e)

υ LL = υ LT = 0.25 ,

where the subscripts of L and T denote the directions parallel and transverse to the fiber directions, respectively. For comparison purposes, the set of normalized variables used in Varadan and Bhaskar [198] is defined as 10 E L

[u x (x, θ , ζ ),

uθ (x, θ , ζ )] =

[σ x (x, θ , ζ ),

σ θ (x, θ , ζ ), τ xθ (x, θ , ζ )] =

q0 h S

3

[u x (x, θ , ζ ),

u θ (x, θ , , ζ )] ,

θr

xr

10 E L q0 h S 4

u r ( x, θ , , ζ ) ,

10 [σ x (x, θ , ζ ), σ θ (x, θ , ζ ), τ xθ (x, θ , ζ )] , q0 S 2

[τ (x, θ , ζ ), τ (x, θ , ζ )] = q10S [τ (x, θ , ζ ), τ (x, θ , ζ )], xr

u r ( x, θ , ζ ) =

θr

0

22

σ r ( x , θ , ζ ) = σ r (x , θ , ζ ) / q 0 ,

(36)

Table 3 shows a set of comparative studies of the results obtained using the RMVT- and PVD-based FCLMs [103, 104], state-space DRK [171], RMVT-based FCPM [133], and meshless DRK collocation and element-free Galerkin methods [117]. The stress and displacement components induced at the crucial positions of the simply-supported, [90 0 / 0 0 / 90 0 ] laminated composite cylinders are presented, in which the relevant parameters of assorted methods are denoted, as follows: N l is the total number of artificially divided layers with an equal thickness for each layer, n

p

is the total number of sampling nodes, uniformly

distributed along the axial direction (or the thickness direction) of the cylinder, n is the highest-order of the base functions, a is the radius of the influence zone for each sampling node, and ∆x = L / (n p − 1) . LM nn

τ

and LD nn

τ u

nσ nw

u

nσ nw

are defined to represent the RMVT- and PVD-based FCLMs, in which nu , nw , nτ and nσ

are the related orders used for expanding the in- and out-of-surface displacement and transverse shear and normal stress components through the thickness coordinate, respectively. It can be seen in Table 3 that the convergent solutions of FCLMs are obtained at Nl=9 and Nl=6 for the quadratic- and cubic-order FCLMs, respectively, and the solutions of RMVT-based FCLMs are more accurate than those of PVD-based ones. Both the solutions of meshless DRK collocation methods based on the strong formulation and those of EFG methods based on the weak formulation closely agree with the exact 3D solutions, with a fast convergence rate. Even though the predictions of meshless DRK collocation methods with regard to the transverse stress components are more accurate than those of EFG methods, and vice versa for the displacement and in-surface stress components, the deviations between them are very minor. The solutions of aforementioned methods converge rapidly, and the convergent solutions are in excellent agreement with the exact 3D [198], modified Pagano [10] and layerwise third-order solutions [199].

7.

Conclusions

The development of various 3D semi-analytical numerical methods available in the literature and their applications to the analyses of laminated composite and multilayered FGEM/FGPM plates/shells with combinations of simply-supported, free and clamped edge conditions was surveyed. Based on the RMVT, the strong and weak formulations of the 3D piezoelectricity theory and their corresponding possible edge boundary conditions were derived and presented. The semi-analytical numerical methods incorporated the state space and asymptotic methods with the DQ, DRK and FE methods, and were classified into two groups, 23

namely, the strong and weak formulation-based semi-analytical numerical methods. The meshless DRK collocation, SaS, state space DQ, state space DRK, asymptotic DQ and asymptotic DRK methods were included in the former, while the finite layer, meshless EFG, finite prism, state space FE and asymptotic FE methods were included in the latter. A comparative study of the results obtained by using assorted semi-analytical numerical methods for the stress and deformation analyses of simply-supported, [900/00/900] laminated circular hollow cylinders under a sinusoidally distributed load was carried out. It is pleasing to note that the quasi-3D solutions obtained from different approaches were found to be consistent. These 3D solutions may thus serve as a standard to assess the assorted 2D theories of plates and shells. The through-thickness distributions of various field variables may serve as a reference for making the appropriate kinetic or kinematics assumptions a priori, when a 2D advanced or refined theory is to be developed. The authors hope this review paper provides a comprehensive overview of the 3D semi-analytical numerical methods available in the existing literature.

Acknowledgments

This work was supported by the Ministry of Science and Technology of the Republic of China through Grant MOST 103-2221-E-006-064-MY3.

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39

(a) ζ A

x A

r

θ

L

Fig. 1(a)

40

(b)

(c)

ζ3

ζ2

ζ1

ζ0

Fig. 1(b)

Fig. 1 (c)

41

Captions: Fig. 1. (a) The configuration and coordinates of a laminated composite or multilayered FGM circular hollow cylinder, (b) lay-up of a [900/00/900] laminated composite cylinder, and (c) lay-up of a sandwiched FGM cylinder

Table 1 Comparisons with regard to the meshes, configuration, interpolation functions, and discretized domains among the present finite cylindrical prism, layerwise shell, and finite cylindrical layer elements [133] Elements Finite cylindrical prism elements Layerwise shell elements Meshes

(the 2x1x2 mesh)

(1/4 cylindrical model with a 2x1x2 mesh)

Finite cylindrical layer elements

(1x1x2 mesh)

Configuration

(a Q8 shell element) (a Q8 prism element)

Interpolation functions

x-direction: Lagrange polynomials θ -direction: Fourier series functions ζ -direction: Lagrange polynomials

Discretized domains

x- ζ surface

(a quadratic layer element) x-direction: Lagrange polynomials θ -direction: Lagrange polynomials ζ -direction: Power series functions x- θ surface

x-direction: Fourier series functions θ -direction: Fourier series functions ζ -direction: Lagrange polynomials ζ -direction

Table 2 Partial list of references on various quasi-3D analyses of laminated composite and multilayered FGEM/FGPM plates and shells using the state space DQ methods Materials and Structures

Bending

Vibration

Mechanical/Piezoelectric

Thermal

Free

Transient

bending

bending

vibration

vibration

Laminated elastic plates

Lu et al. [139 ]

Chen and Lue [136]

and shells

Lu et al. [142]

Lu et al. [137]

Alibeigloo and Shakeri [145]

Alibeigloo and Shakeri [138]

Alibeigloo [146]

Lu et al. [141] Alibeigloo [146]

FGEM plates and shells

Lu et al. [143]

Ying et al. [147]

Nie and Zhong [140]

Liang et al. [160]

Alibeigloo [154]

Alibeigloo [170]

Nie and Zhong [150]

Liang et al. [164]

Zhang et al. [162]

Jodaei et al. [153]

Liang et al. [165]

Alibeigloo and Alizadeh [163]

Jodaei and Yas [155]

Liang et al. [168]

Alibeigloo and Emtehani [166]

Kermani et al. [158] Jodaei et al. [159] Alibeigloo and Alizadeh [163] Alibeigloo and Emtehani [166]

Laminated piezoelectric plates

Zhou et al. [148]

Zhou et al. [144]

And shells

Feri et al. [167]

Alibeigloo and Kani [149] Feri et al. [167]

FGPM plates and shells

Alibeigloo and Nouri [151]

Alibeigloo et al. [156]

Alibeigloo and Simintan [152]

Yas et al. [157]

Jodaei [161]

Yas and Moloudi [169]

Table 3 Comparative studies of the results obtained by using various semi-analytical numerical methods for the stress and deformation analyses of the [90° / 0° / 90°] laminated composite cylinder with fully simple supports and under a sinusoidally distributed load (L/R=4 and R/h=4). Related  L h  π h L  L π  L   L h σ x  ,0,  σ θ  ,0,  τ xθ  0, ,  σ r  ,0,0  τ θ r  , ,0  ur  , 0, 0  Theories 2 2 2 2 2 2 2 2 2 2             parameters

(Nl=3) LM22 22

RMVT-based FCLMs [103]

0.1148

6.4227

0.1070

-0.5858

-2.1171

3.9792

(Nl=6)

0.1238

6.5330

0 . 10 80

-0.6370

-2.4628

4.0055

(Nl=9)

0.1256

6.5412

0.1081

-0.6152

-2.3223

4.0087

(Nl=3) LM33 33

0.1292

6.5462

0.1081

-0.6194

-2.3489

4.0082

LM

22 22

LM

22 22

33 33

(Nl=6) (Nl=9)

0.1267

6.5442

0.1081

-0.6194

-2.3898

4.0090

33 33

0.1271

6.5446

0.1081

-0.6194

-2.3488

4.0090

22 22

(Nl=3)

0.1158

6.3474

0.1062

-0.6117

-2.2939

3.9485

22 22 22 22

(Nl=6) (Nl=9) (Nl=3) (Nl=6) (Nl=9)

0.1256 0.1266

6.5289 6.5411

0.1079 0.1081

-0.6188 -0.6192

-2.3448 -2.3480

4.0042 4.0080

0.1282

6.5441

0.1081

-0.6193

-2.3488

4.0082

0.1271

6.5446

0.1081

-0.6194

-2.3488

4.0089

n p= 7

0.1270 0.1273

6.5446 6.5409

0.1081 NA

-0.6194 -0.6194

-2.3488 NA

4.0089 4.0134

LM LM

PVD-based FCLMs [104]

LD LD LD

LD33 33 LD33 33 33 33

LD

State space DRK method (n=4, a=4.1Δx, Nl=9) [171]

RMVT-based Q8 FCPM [133]

RMVT-based Q9 FCPM [133]

n p= 9

0.1267

6.5385

NA

-0.6196

NA

4.0113

np=11

0.1267

6.5384

NA

-0.6196

NA

4.0113

np=21 Mesh (36x6)

0.1269 0.1238

6.5383 6.5335

NA 0.1080

-0.6195 -0.8307

NA -2.6386

4.0113 4.0056

Mesh (48x9)

0.1260

6.5415

0.1081

-0.8221

-2.5761

4.0088

Mesh (48x12)

0.1262

6.5433

0.1081

-0.8189

-2.5533

4.0086

Mesh (36x6) Mesh (48x9)

0.1238 0.1256

6.5333 6.5414

0.1080 0.1081

-0.8307 -0.8221

-2.6387 -2.5761

4.0056 4.0088

Mesh (48x12)

0.1262

6.5432

0.1081

-0.8189

-2.5533

4.0086

Meshless DRK collocation

n p= 5

0.1273

6.5629

0.1083

-0.6200

-2.3524

4.0157

(n=3, a=3.1Δx, Nl=3) [117]

n p= 7 n p= 9

0.1271 0.1270

6.5480 6.5456

0.1081 0.1081

-0.6195 -0.6194

-2.3496 -2.3491

4.0102 4.0093

np=11

0.1270

6.5450

0.1081

-0.6194

-2.3489

4.0091

np=21

0.1270

6.5446

0.1081

-0.6194

-2.3488

4.0090

n p= 5 n p= 7

0.1258 0.1265

6.5427 6.5438

0.1081 0.1081

-0.6219 -0.6174

-2.3587 -2.3422

4.0086 4.0091

n p= 9

0.1267

6.5441

0.1081

-0.6210

-2.3547

4.0089

np=11

0.1268

6.5442

0.1081

-0.6182

-2.3451

4.0090

np=21 Modified Pagano method [10]

0.1269 0.1269

6.5444 6.5462

0.1081 0 . 10 82

-0.6198 -0.6192

-2.3504 -2.348

4.0089 4.0097

Layerwise third-order theory [199]

0.1217

6.533

0.1080

-0.62

-2.346

4.006

3D elasticity solutions [198]

0.1270

6.545

0.1081

-0.62

-2.349

4.009

Element-Free Galerkin (n=3, a=3.1Δx, Nl=3) [117]