CC BY
39
DOI: 10.17277/amt.2016.01.pp.061-066
Sampling Surfaces Formulation for Functionally Graded and Laminated Composite Shells
G.M. Kulikov*, A.A. Mamontov, S.V. Plotnikova, M.G. Kulikov, S.A. Mamontov
Department of Applied Mathematics and Mechanics, Tambov State Technical University, 106, Sovetskaya, Tambov, 392000, Russian Federation
* Corresponding author. Tel.: + 7 (4752) 63 04 41. E-mail: [email protected]
Abstract
This paper focuses on a finite element implementation of the sampling surfaces (SaS) method for the three-dimensional (3D) stress analysis of functionally graded (FG) laminated elastic and electroelastic shells. The SaS formulation is based on choosing inside the nth layer In not equally spaced SaS parallel to the middle surface of the shell in order to introduce the electric potentials and displacements of these surfaces as basic shell variables. Such choice of unknowns permits the presentation of the proposed FG shell formulation in a very compact form. The SaS are located inside each layer at Chebyshev polynomial nodes that improves the convergence of the SaS method significantly.
Keywords
Functionally graded material; laminated piezoelectric shell; sampling surfaces method, exact geometry solid-shell element.
© G.M. Kulikov, A.A. Mamontov, S.V. Plotnikova, M.G. Kulikov, S.A. Mamontov, 2016
Introduction
The exact 3D analysis of FG piezoelectric shells has received considerable attention during past ten years. There are at least five approaches to the exact 3D analysis of electroelasticity for FG piezoelectric laminated shells, namely, the modified Pagano approach, the series expansion approach, the state space approach, the asymptotic approach and the SaS approach. The first four approaches were applied efficiently to the 3D analysis of FG piezoelectric structures in many papers [1]. Recently, the SaS approach has been also applied to the 3D modelling of FG piezoelectric plates and shells [2-4].
In accordance with the SaS method, we choose inside the nth layer In not equally spaced SaS
Q(n)1, Q(n)2,..., Q(n)In parallel to the middle surface and introduce displacement vectors u(n)1, u(n)2,..., u(n)In and electric potentials 9(n)1, 9(n)2,..., 9(n)In of these surfaces as basic shell
variables, where In > 3 . Such choice of displacements and electric potentials with the consequent use of Lagrange polynomials of degree In -1 in the thickness direction for each layer permits the presentation of governing equations of the FG piezoelectric laminated shell formulation in a very compact form.
It is worth noting that the developed approach with equally spaced SaS does not work properly with Lagrange polynomials of high degree because of the Runge's phenomenon, which yields the wild oscillation at the edges of the interval when the user deals with any specific functions. If the number of equally spaced nodes is increased then the oscillations become even larger. However, the use of Chebyshev polynomial nodes allows one to minimize uniformly the error due to Lagrange interpolation. This fact gives an opportunity to find solutions of the 3D static problems of elasticity and electroelasticity for FG laminated shells with a prescribed accuracy employing the sufficiently large number of SaS.
SaS Formulation for Displacement and Strain Fields
Consider a thick laminated shell of the thickness h. Let the middle surface Q be described by orthogonal curvilinear coordinates and 02, which are referred to the lines of principal curvatures of its surface. The coordinate 03 is oriented along the unit vector e3 normal to the middle surface. Introduce the following notations: ea are the orthonormal base vectors of the middle surface; Aa are the coefficients of the first fundamental form; ka are the principal
curvatures of the middle surface; c\
(n)l" = 1 + ka0(3n)l"
are the components of the shifter tensor at SaS; 03n)i"
are the transverse coordinates of SaS inside the nth layer given by:
0W1 =0["-1]5
e(") In =0[«],
0( n)mn _
:(e(n-1] +e(n] )- 1 K
cos
. 2mn - 3
'2(In - 2)
(1)
(2)
where 03n 1] and 03n] are the transverse coordinates of
layer interfaces and depicted in Fig. 1;
hn = 03n] -03n-1] is the thickness of the nth layer. Here
and in the following developments, the index n identifies the belonging of any quantity to the nth layer and runs from 1 to N, where N is the number of layers; the index mn identifies the belonging of any quantity to the inner SaS of the nth layer and runs from 2 to In -1, whereas the indices in , jn, kn to be used later describe all SaS of the nth layer and run from 1 to In ;
Greek indices a, p range from 1 to 2; Latin tensorial indices i, j, k, l range from 1 to 3.
It is seen from (2) that transverse coordinates of inner SaS of the nth layer coincide with coordinates of the Chebyshev polynomial nodes [5]. This fact has a great meaning for a convergence of the SaS method [6].
The strain tensor at SaS of the nth layer in a middle surface frame ei can be written as follows [6]:
2b'
( n)in
1
-A
( n)in
1
X
( n)in
aß c(n)in aß c(n)in ßa '
Cß Ca
2B(n)in _ß(n)in + 1 A(n)in B(n)in =R(n)in (3)
Zfca3 _ Pa c(n)^ A3a ' fc33 " P3 ' W
where A(;a)in are the strain parameters of SaS defined as:
Aaan _ 1 ua,an + ^a^ß )n + )n
^a
Aß?n _ -1-ußnain - Bauan)in for ß ^ a,
Aa
A(n)in _ 1 u (n)in - k u (n)in 3a " . 3,a Kaua Aa
^a1
1a,ß
for ß ^ a,
(4)
where M(n) i n = ui (03n)i n) are the displacements of SaS; p(n)in = ui,3(03n)in) are the derivatives of
displacements with respect to thickness coordinate at SaS.
Now, we start with the first assumption of the proposed piezoelectric laminated shell formulation. Let
us assume that displacements of the nth layer u(n) are
distributed through the thickness as follows:
,(n)
_ £L(n)inu(n)in, e3n-1] < e3 < e3n], (5)
where L{n)l" (03) are the Lagrange polynomials of degree In -1 expressed as
e e(n)Jn
L(n)in _ T-f e3 - e3
n
Fig. 1. Geometry of the laminated shell
jn *'n 3
e(n)in -e(n)jn '
(6)
1
n
Using equations (5) and (6), one obtains
p:
( n)in
£M(n)Jn (03n)'n )n) jn
(7)
where M(n)Jn = LnJn are the derivatives of the Lagrange polynomials. It is seen that the key functions P(n)in of the laminated shell formulation are represented according to (7) as a linear combination of displacements of SaS of the nth layer u(n)Jn .
The following step consists in a choice of the suitable approximation of strains through the thickness of the nth layer. It is apparent that the strain distribution should be chosen similar to the displacement distribution (5). Thus, the second assumption of the developed shell formulation can be written as
,(n) _
= £L
(n)in o(n)in
03n-1] <03 <03n].
■'3 ^03 <03 . (8)
The strain-displacement relationships (3) and (8) exactly represent all rigid-body motions of the laminated shell in any convected curvilinear coordinate system. The proof of this statement is given in paper [6].
SaS Formulation for Electric Field
The relation between the electric field Et and the electric potential 9 is given by
Ea =--
1
Aa (1 + W
9, a, E3 = -9v (9)
In particular, the electric field at SaS of the nth layer Ei(n)in = E, ^^) is presented ;
as
(n)in _
ZT(")* n a
__1_
A c(n)in
E
(n)in (n)i
= -v
(10)
where 9(n)in = 9^n)in) are the electric potentials of
SaS of the nth layer; y(n)in = 93(03n)<n) are the values
of the derivative of the electric potential with respect to thickness coordinate on SaS.
Next, we accept the third and fourth assumptions of the proposed piezoelectric laminated shell formulation. Let the electric potential and the electric field be distributed through the thickness of the nth layer as follows [3]:
(n)in
9(n) = £L(n)in 9(n)in, 03n-1] < 03 < 03n],
(11)
E(n) = £L^nE^n, 03n-1] <03 <03n]. (12)
The use of (6) and (11) yields a simple formula
yWn = £M(n)Jn ^^n )9(n)Jn , (13)
Jn
which is similar to (7). This implies that the key functions y(n)in of the piezoelectric laminated shell formulation are represented as a linear combination of electric potentials of SaS of the nth layer 9(n) J .
Variational Formulation
The variational equation for the piezoelectric laminated shell in the case of conservative loading can be written as
5n = 0, (14)
where n is the extended potential energy defined as
n =
03
2ff£ i (( - D(n)E(n)))
Q n 0[n-l]
x(1 + k103)(1 + k203)d01d02d03 - W, (15)
where c(Jn) is the stress tensor of the nth layer; Dis
the electric displacement vector of the nth layer; W is the work done by external electromechanical loads.
Substituting strain and electric field distributions (8) and (12) in the extended potential energy (15) and introducing stress resultants and electric displacement resultants
H\p,n = fa¡/n)L(n)'n (1 + k103)(1 + k203)d03, (16)
0[ n-1]
03
e3n]
T(n)in = f D(n) L(n)in (1 + k103)(1 + k203)d03, (17)
0[n-1]
3
one finds
n =
2 if££('
H(n)in8(")in - T(n)inE(n)in )X
2 Q n in
x A A
- W.
(18)
n
n
n
n
n]
0
n
For simplicity, we consider the case of linear piezoelectric materials
4n) = Cjl^tf - jkn), 03n-1] < 03 < 03n], (19)
^ki^k
D(n) = } + Ekn\ 03n-1] < 03 < 03n], (20)
where C(jkkl, ekn and are the elastic, piezoelectric
and dielectric constants of the nth layer.
Now, we accept the fifth and last assumption of the FG piezoelectric laminated shell formulation. Let us assume that the material constants are distributed through the thickness of the shell according to the following law
(n)inC(n)in e(n) = V T(n)ine(n)in
C(n) = V T(n)inC(n)in n) - V T( ^ijkl ~ Zj^ ijkl > ekij
^kij
-(n)= V T(n)in Ân)'n ik ik
(21)
that is extensively utilized in this paper, where C'"^'"
ijkl
ekj
and e(n)<n are the values of elastic, piezoelectric
and dielectric constants on SaS of the nth layer.
Finite Element Formulation
The variational equation (14) and (18) is the basis for developing the exact geometry (EG) four-node solid-shell element proposed in papers [7, 8]. The term EG reflects the fact that the parametrization of the middle surface is known and, therefore, coefficients of the first and second fundamental forms and Christoffel symbols are taken exactly at element nodes. In the EG shell element formulation, the displacement and electric field vectors of SaS of the nth layer are resolved in a surface frame ei . This in turn allows the implementation of the efficient analytical integration inside the EG solid-shell element.
The finite element formulation is based on the simple interpolation of the shell via curved EG four-node solid-shell elements
(n)in
= 1 N
u (n)in ruir ■
(n)in
(22)
where ^a=(0a- ca)/1 a are the normalized curvilinear coordinates (see Fig. 2); Nr (^1, ) are the bilinear shape functions of the element;
ur''n - uy'n (Pr ) and
tfVn = u(n)in 1
-9(n)in ( ) are the
values of displacements and electric potentials of SaS at element nodes Pr in (^1, ) -space; the index r runs from 1 to 4 and denotes the number of nodes.
To implement the analytical integration throughout the EG shell element [7], we employ the assumed interpolation of strain and electric field components
s
(n)in
= 1N
s(n)in r&ijr
s
(n)in (n
ijr
sfn (Pr ), (23)
E"" = X NrE(rn)'", E^'" = E(n)'" (pr). (24)
r
Note that the main idea of such approach can be traced back to the ANS method (see, e.g. [9]). In contrast with the conventional ANS formulation, we treat the term "ANS method" in a broader sense. In our EG piezoelectric solid-shell element formulation, all strain and electric field components are assumed to vary bilinearly throughout the element. This implies that instead of the expected non-linear interpolation of strain and electric field components in accordance with equations (3), (4) and (10) the more suitable bilinear ANS interpolation is utilized. It is important that we advocate the use of the extended ANS method (23) and (24) to implement the efficient analytical integration inside the EG shell element.
r
n
r
r
Numerical Examples
1. First, we study a simply supported cylindrical shell with L/R = 4 subjected to the sinusoidally
distributed transverse load a^ =- ^sin cos492 at
the bottom surface, where 01 and 02 are the longitudinal and circumferential coordinates of the middle surface; L and R are the length and radius of the shell. The shell is made of the unidirectional composite with the fibers oriented in the circumferential direction. The mechanical parameters
G
[0] =g0] =gÍ0] =q,f0] =g[3] =g|3] =9[3] =0,
are taken as EL = 25ET.
Glt = 0.5Et, Gtt = 0.2Et,
ET =10°,
Vlt = vtt = 0.25 . Here, subscripts L and T refer to the fiber and transverse directions of the layer. To compare the derived results with the 3D exact solution [10], the following dimensionless variables are utilized:
u3 = 10El h3«3(L/2, 0, z)/R4p0, G22 = 10h2g22(L /2, 0, z)/ R2p0,
Qi3 = 100haB(0, 0, z)/ Rpo, c23 = 10ha23(L/2, n/8, z)/Rp0, z = 03/h.
Due to symmetry of the problem, only one sixteenth of the shell is discretized using the 32 x128 mesh of EG four-node solid-shell elements. The data listed in Table 1 demonstrate the high potential of the developed SaS finite element formulation, which provides three right digits for basic variables at crucial points utilizing seven SaS inside the moderately thick and thin shells.
2. Consider next a symmetric piezoelectric three-layer cylindrical shell with equal ply thicknesses subjected to mechanical loading acting on the top surface
g333] = p0 sin
n01 L
cos 02
or electric loading acting on the same surface
c[0] =40]=4°3]=40] =4] =433=4]=0,
D[3] = q0 sinn01cos02,
where p0 = -1 Pa and q0 = 1 C/m2 . The bottom and top layers are composed of the FG piezoelectric material, whereas the central layer is made of the homogeneous piezoelectric material. It is assumed that the FG material properties are distributed in the thickness direction according to the exponential law [1]
C(1) = C(2)ew(z) e(1) = e(2)em(z)
ijkl ~ ijkl ' ikl ~ ikl
e(1)=e(2)ew(z), -1/2 < z <-1/6,
C (3) = C (2)e^3 ( z) „(3) = „(2)e^3 ( z) ijkl ~ ijkl ' eikl ~ ikl c
c(3) = c(2) eMz)
ik
e™-', 1/6 < z < 1/2, z) = -a(6z + 1)/2, ^,3( z) = a(6z -1)/2,
z = 03/h,
where a is the material gradient index; C(k], e\u and are the elastic, piezoelectric and dielectric
=(2)
^ik
constants of the central layer, which are considered to be the same as those of the PZT-4, whose material properties are given in [1, 4].
The geometric parameters of the shell are taken to be L = 4 m and R = 1m. To compare the results derived with the analytical solution [1], we introduce
Table 1
Results for the composite cylindrical shell using seven SaS and 32 x128 mesh
R / h
EG solid-shell element formulation
Varadan and Bhaskar [10]
«3(0) G22 (0.5) G13 ( 0) G23 ( 0 ) «3(0) G22 (0.5) G13 ( 0 ) G23 ( 0 )
4 2.782 4.854 0.9863 -2.970 2.783 4.859 0.987 -2.990
10 0.9188 4.048 0.5199 -3.665 0.9189 4.051 0.520 -3.669
100 0.5169 3.840 0.3927 -3.856 0.5170 3.843 0.393 -3.859
Fig. 3. Through-thickness distribution of transverse shear stresses of the FG three-layer cylindrical shell under mechanical loading:
EG finite element formulation (—) and analytical solution [1] (O)
Fig. 4. Through-thickness distribution of transverse shear stresses of the FG three-layer cylindrical shell under electric loading:
EG finite element formulation (—) and analytical solution [1] (O)
dimensionless variables c13 = c13 (0, 0, z) / p0 in the case of mechanical loading and a13 = 10-6 x xc13(0, 0, z)q */p * q0 in the case of electric loading,
2 2 where p* = 1 N/m , q* = 1 C/m and z = 03 / h .
Figures 3 and 4 display the distribution of transverse shear stresses in the thickness direction for different values of the slenderness ratio R/h and the material gradient index a employing nine SaS for each layer and the 64 x 64 uniform mesh of EG four-node solid-shell elements. These results demonstrate again the high potential of the proposed SaS solid-shell element formulation. This is due to the fact that boundary conditions on the bottom and top surfaces and continuity conditions at interfaces for transverse shear stresses are satisfied exactly.
Conclusions
The SaS formulation for the 3D analysis of FG piezoelectric laminated shells has been developed. This formulation is based on choosing the SaS located
at Chebyshev polynomial nodes throughout the layers. Such choice permits one to minimize uniformly the error due to Lagrange interpolation. The SaS formulation for piezoelectric laminated shells is based on 3D constitutive equations and gives the possibility to obtain 3D solutions for FG piezoelectric shells with a prescribed accuracy, which asymptotically approach the exact solutions of piezoelectricity as the number of SaS goes to infinity.
References
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