CC BY
35
DOI: 10.17277/amt.2017.01.pp.044-055
Hybrid-Mixed Quadrilateral Element for Laminated Plates Composed of Functionally Graded Materials
G.M. Kulikov1*, E. Carrera2, S.V. Plotnikova1
1 Laboratory of Intelligent Materials and Structures, Tambov State Technical University, 106, Sovetskaya St., Tambov, 392000, Russia 2 Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, 24, Corso Duca degli Abruzzi, Turin, 10129, Italy
* Corresponding author. Tel.: + 7 (4752) 63 04 41. E-mail: [email protected]
Abstract
A hybrid-mixed four-node quadrilateral element for the three-dimensional stress analysis of laminated functionally graded plates through the use of the method of sampling surfaces is developed. The sampling surfaces formulation is based on choosing inside the nh layer In not equally spaced sampling surfaces parallel to the middle surface, in order to introduce the displacements of these surfaces as basic plate variables. The sampling surfaces are located inside each layer at Chebyshev polynomial nodes that allows one to uniformly minimize the error due to the Lagrange interpolation. To avoid shear locking and have no spurious zero energy modes, the assumed natural strain concept is employed. The developed assumed natural strain four-node quadrilateral plate element passes the bending patch tests for laminated and functionally graded plates and exhibits a superior performance in the case of coarse distorted mesh configurations. It can be useful for the three-dimensional stress analysis of thick and thin laminated functionally graded plates because the sampling surfaces formulation gives the possibility to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach the three-dimensional exact solutions of elasticity as the number of sampling surfaces tends to infinity.
Keywords
Functionally graded material; hybrid-mixed quadrilateral element; laminated plate; sampling surfaces formulation; three-dimensional stress analysis.
© G.M. Kulikov, E. Carrera, S.V. Plotnikova, 2017
Introduction
The functionally graded (FG) materials are a new class of advanced materials in which the material properties vary continuously from point to point. This property is achieved by varying the volume fraction of constituents. The FG materials are usually made by mixing the metal and ceramic phases. The ceramic with the low thermal conductivity serves as a thermal barrier and is placed at high temperature locations, whereas the metal is placed at regions where the mechanical properties such as toughness need to be high. The concept of the FG material was proposed by Japanese material scientists in the 1980s. However, most of the publications in the open literature appeared in the last fifteen years. The progress in the analytical
and numerical modelling and analysis of FG materials and structures is reviewed in [1-3].
At present, the finite element method is widely used in bending, buckling and vibration analyses of FG plates and shells because of its advantages compared with other numerical techniques. The finite element formulations for the FG plates were developed in [4-7] through the first-order shear deformation theory [4] also known as a Reissner-Mindlin plate theory [5]. The major problem in constructing the low-order triangular and quadrilateral plate elements is how to eliminate shear locking for thin plates. The nodal integrated triangular and quadrilateral finite element formulation is presented in [6]. The proposed plate elements are free of locking and show little sensitivity to geometric distortions. The isogeometric finite
element formulation based on non-uniform rational B-splines (NURBS) was considered in [7]. It was established that shear locking can be overcome employing the NURBS functions of high order.
The finite element model for FG plates using the higher-order shear deformation theory was proposed by Reddy [8]. More general finite element formulations accounting for the transverse normal deformation for bending, buckling and free vibration analyses of FG plates are presented in [9, 10]. The nine-node rectangular plate element with 13 degrees of freedom (DOFs) per node has been developed in [9]. The NURBS-based finite element method for FG homogeneous and sandwich plates has been proposed in [10]. The feature of this approach is that only four displacement DOFs are utilized. The ^-continuity requirement, however, can be carried out with the help of NURBS functions and no shear locking occurs.
Note that all finite elements considered do not describe well the through-thickness distribution of transverse stresses except for [11], where the efficient nine-node rectangular FG shell element with 15 DOFs per node has been developed via Carrera's fourth-order shell formulation [12, 13]. The shear and membrane locking phenomena are prevented through the assumed natural strain (ANS) method. However, the authors report that the transverse normal stress is calculated with a large error in the case of thin plates because of the small value of this stress compared to in-plane stresses.
The present paper is intended to overcome the above mentioned difficulties and develop the finite element that makes it possible to evaluate all stress components effectively for the thick and very thin FG plates. To solve such a problem, the four-node quadrilateral plate element using the sampling surfaces (SaS) technique [14, 15] is proposed. The SaS formulation is based on choosing inside the nth layer the arbitrary number of surfaces Q(n)1, Q(n)2,..., Q(n)In parallel to the middle surface in order to introduce the displacement vectors u(n)1, u(n)2,..., u(ny" of these surfaces as basic plate variables, where In is the number of SaS of the nth layer (In > 3). Such a choice of unknowns with the use of Lagrange polynomials of degree In - 1 in the assumed approximations of displacements, strains and material properties through the layer thicknesses leads to a very compact form of governing equations of the SaS laminated FG plate formulation. Recently, the SaS formulation has been utilized by Kulikov and Plotnikova to evaluate analytically the three-dimensional (3D) stress state in FG plates and shells [16-19].
In a proposed hybrid-mixed four-node finite element formulation, all SaS are located at Chebyshev polynomial nodes throughout the layers that make it possible to use the Lagrange polynomials of high degree [20]. To avoid shear locking and have no spurious zero energy modes in the case of distorted meshes, the assumed interpolations of displacement-independent strains and stress resultants in conjunction with the ANS interpolation of displacement-dependent transverse shear strains are utilized. To solve this problem, the Hu-Washizu multivariable variational principle is invoked. The developed isoparametric hybrid stress-strain finite element for the 3D stress analysis of laminated FG plates has computational advantages compared to the hybrid strain and hybrid stress finite elements. This is due to the fact that here no expensive numerical inversion of elemental matrices is needed; all matrix inversions can be done analytically. On the contrary, in conventional hybrid-mixed finite element formulations [21, 22] the inversion of the flexibility matrix is required. In fact, it is the most costly operation because the number of stress or strain parameters (modes) to be introduced to analyze effectively the laminated FG plates can be sufficiently large [23]. Note also that the proposed hybrid-mixed finite element model for the 3D stress analysis of laminated FG plates generalizes the authors' displacement-based finite element models [24, 25] and the hybrid-mixed quadrilateral four-node element models developed for single-layer and multilayered plates [26, 27] as well.
Three-Dimensional Description of Laminated Plate
Consider a laminated plate of the thickness h composed of N inhomogeneous layers. Let the middle surface Q be described by Cartesian coordinates xl and x2, whereas the coordinate x3 is oriented in the thickness direction. The transverse coordinates of SaS of the nth layer x3n)in are defined as
(n)1 _ x[n-1].
x-
(n)In _ x[n]
r(n)mn -
(x3 2 3
[n-1] + x[n]
3 ) - 2 hncos
2mn - 3
:---
2(In - 2)
(1) , (2)
where x3n-1] and x3n] are the transverse coordinates of layer interfaces Q[n - 1] and Q[n] depicted in Fig. 1; hn _ x3n] - x3n-1] is the thickness of the nth layer; In is the number of SaS corresponding to the nth layer; the layer index n runs from 1 to N; the index mn identifies the belonging of any quantity to the inner SaS of the nth layer and runs from 2 to In - 1; the indices in, jn, kn are introduced for describing all SaS of the nth layer and run from 1 to In; the Latin indices i, j, k, l range from 1 to 3; the Greek indices a, p range from 1 to 2.
Fig. 1. Geometry of the laminated plate
It is important that the transverse coordinates of inner SaS (2) coincide with the coordinates of Chebyshev polynomial nodes [20]. This fact has a great meaning for the convergence characteristics of the SaS method [16-19].
The strains of the nth layer s(n) can be written as
2s(,n) = u(j + u (n)
(3)
"¡j ¡,j j,i '
where u(n) are the displacements of the nth layer.
Let us introduce the first two assumptions of the SaS laminated plate formulation. Assume that the displacements and strains are distributed through the thickness of the nth layer as follows:
,(n) =
= £L
(n)'nu(n)'n
u
(n)'n = u(n)(x^n ),
(4)
(n) =
= £ L
(n)'n s(n)in .
s
(n)in =s(n)( x(n)'
= s(")('n ), (5)
where u(n)-n (x1, x2) are the displacements of SaS of the nth layer Q(n)-n; s(/n)-n (x1, x2) are the strains of the
same SaS; L(n)-n (x3) are the Lagrange polynomials of degree In - 1 defined as
L
(n)in -
n
Xn X'
(n) jn
jn *'n x3
(n)in - Xv>Jn
x(n) jn
l3
(6)
The use of (3)-(6) leads to relations between the SaS displacements and SaS strains:
2s(n)in = u(n)in + u(n)in . ap = ua,p +
p,a
2s(n)in = R(n)in + u (n)'n-
2Sa3 =Pa +u
s3n/n =p3n)in, (7)
where p(n)-n (x1, x2) are the values of the derivatives of displacements with respect to the thickness coordinate at SaS given by
P(n)in = M.,3 (x3n)in ) = £M(n) jn (x3n)in )u(n) jn , (8)
where M(n)Jn = Jn are the derivatives of Lagrange polynomials, which are calculated at SaS as
M
(n) j,
'(x3n)in ) =
Sn) jn
- X
(n)in
n
(n)in
- X
(n)kn
kn , jn X3
(n) jn
- X
(n)kn
for jn * ¡n,
M (n)'n (x3n)i'n )= - £ M(n) jn (( ).
(9)
jn ^n
It is seen from (8) that the key functions p(n)-n of the SaS laminated plate formulation are represented as a linear combination of displacements of SaS of the nth layer u(n)Jn .
Hu-Washizu Mixed Variational Formulation for Laminated FG Plate
To develop the assumed stress-strain finite element formulation, we have to invoke the Hu-Washizu multi-field variational principle in which displacements, strains and stresses are utilized as independent variables [28]. It can be written as follows:
(10)
jhw = 0
x[n]
Jhw =j]l j
a n X[n-1] X3
1 e(n)c(n)e(n) -
2 ij ijklkl '
-CT
j (e(n) -j )^dx1dx2dx3 - W, W = jj(p+u+ - p~u~ )dx1dx2 + WS
(11)
where CT(n) are the stresses of the nth layer; Cjl are
the elastic constants of the nm layer; e(n) are the displacement-independent strains of the nth layer;
are the displacements
u- = u,(1)1
and u+ = u
+ _,,( N )IN
of bottom and top surfaces; p- and p+ are the tractions acting on the bottom and top surfaces; WE is the work done by external loads applied to the edge surface E. Here and in the following developments, the summation on repeated indices is implied.
Following the SaS technique, we introduce the third assumption of the proposed hybrid-stress finite element formulation. Let the displacement-independent strains be distributed through the thickness similar to displacement and displacement-dependent strain distributions (4) and (5), that is,
T(n)'ne(n)'n .
e(n)in = e(n)
j (x3n)in ), (12)
where e
(n)in
„j '( x1, x2) are the displacement-independent strains of SaS of the nth layer.
1
n
a
n
n
n
Next, we introduce the last assumption of the SaS laminated FG plate formulation. Let the material constants of the nth layer be distributed through the plate thickness as:
C(") - V T(n)inr(n)in • ijkl ~ ^ ijkl ■
r (n)in - r (n)( ijkl ~ ijkl v
r (n)in
ijkl y X3
), (13)
the displacement vector are approximated according to the standard °C interpolation:
-I N
-I Nru(n)ln ;
(17)
(18)
where Cjl'n (x1, x2) are the values of material
constants of the nth layer on SaS.
Substituting the through-thickness distributions (5), (12) and (13) into the Hu-Washizu variational principle (10) and (11), and introducing stress resultants
[n]
Hfn - Jc(n) L(n)'ndx3, J.n-1]
(14)
the following variational equation is obtained:
jji i
Q n 'n
)(e(n)in )TI H(n)in -II \(n)injnk, V jn k,
- IIl(n)injnkn C(n) jne(n)kn
- ô(H(n)in )T (e(n)in - g(n)in ) - ô(s(n)in )T H(n)in
- jj(ôu+ - p-Su-)dx1dx2 + ôWE - 0,
dx1dx2 +
Q
where
8(n)in -
(15)
8
■ (n)in e(n)in e(n)in
'11
'22
J33
2p(n)in
e
H (n)in
(n)in _
(n)in (n)in (n)in
'11
H (n)in H11
22
H (n)in H 22
'33
2e(n)in 12
9p(n)i«
2fc13
2e(n)in
13
9P(n)in
23
2e(n)in
23
H(n)in
H 33
H(n)in H12
H(n)in
H13
H(n)in
H 23
C
(n)jn -
r
(n) jn 1111
r
(n) jn 1122
r
(n) jn
1133
r
(n) jn 1112
r,(n)jn 2211
r(n)jn 3311
r(n)jn 1211
0 0
r,(n)jn 2222
r(n)jn 3322
r(n)jn 1222
0 0
r,(n)jn
2233
r (n) jn 3333
r (n) jn 1233
0 0
(n) jn 2212 r(n) jn
3312
r(n) jn 1212
0 0
0 0 0 0
r(n)jn
1313
0 0 0 0
r (n) jn 1323
r
(n) jn
2313
r
(n) jn
2323 J
Nr _ 4(1 + nlri1 )(1 + n2ri2), (19)
where Nr (i1, i2) are the bilinear shape functions of the finite element with n11 _ n14 _ n21 _ n22 _ 1 and n12 _ n13 _ n23 _ n24 _ -1; xa are the nodal coordinates; are the displacements of SaS
at element nodes; the index r denotes the number of nodes and ranges from 1 to 4. The surface traction vector is also assumed to vary bilinearly throughout the element. The local numbering of the corner nodes and middle side nodes is shown in Fig. 2.
To overcome shear locking and have no spurious zero energy modes, the robust ANS interpolation [29, 30] of the transverse shear strains of SaS with four sampling points can be employed:
P(n)in - ¡/P 8(n)in • a3 abp3 '
(20)
s(3)in _i(1 -i2)s(n)in(B) + i(1 + i2)n(D); (21) eg*" _2(1 -i1 )s2n3)in(A) + 2(1 + i1 ))in(C), (22)
where are the covariant components of the strain tensor of SaS in the contravariant basis aa _ Ijpkp, a3 _ k 3, which are defined by the orthogonality
'
condition aiaJ _5j (see Fig. 2).
The transverse shear strains at sampling
points A, B, C and D are evaluated according to (7), (8) and (18) as:
x:
n]
A(
(n)injnkn -
n - J L(n),nL(n)JnL(n)kndx3. (16) x[n-1]
Hybrid-Mixed ANS Finite Element Formulation
In the isoparametric four-node plate element formulation (from this point, Cartesian coordinates are denoted
12 3
by x , x , x ), the position vector and
Fig. 2. Quadrilateral plate element
x
r
u
n
r
+
2s(fn (B) = i ( - )+
+ 4(n)jn (z(n)'n ( - X3a)(jn + u^n );
in
2s(f" (D) = i(" - u32)i" )+
+ 4 SM (n) ^ ( - + uOf" ),;
in
282"/" (A) = I (- - u33)i" )+
+1(n)jn (z(-)i- )a - x3a)^2) j- + uS3) jn);
282f-(C) = 2 (" - u34)i- )+ +1 ZM(-)j- (z(n)'n )a- jn + ua-4)j- ), (23)
j n
where z(n)'n stands for the transverse coordinates of SaS of the nth layer O(-)l- .
Substituting (18) in strain-displacement equations (7) and (8), and using the ANS interpolation (21)-(23) and presentation for the derivatives of shape functions
~dNr' ~dNr'
cX1 dNr — J-1 df1 dNr
_cX2 _ 2 _
(24)
one obtains
r.(-)i- _ T>(n)i
£v"— Bv U, (25)
where J-1 is the inverse Jacobian matrix;
B
(-)i- (s1, ^2) are the strain-displacement transformation matrices of order 6x12 NSaS;
NSaS — ^I" - N +1 is the total number of SaS; U
-
is the element displacement vector given by:
T
U =
[uT uT uT uT
U r =
(u[f (1)2)T (u!)T (u?)2)T...
...((n-! -1 )T (u[N-1])T(u(N)2)T -1 )T ([n])
[m] [ [m] [m] [m]lT ( n i a A
ur J — \u[/ 4rJ u3^ (m — 0,1,...,N),
,(-)m- =
,(n)m- u (-)m- u (-)m-
(( = 2,...,I- -1).
(26)
To improve the computational efficiency of the ANS four-node quadrilateral plate element, a hybrid-mixed method may be employed. In order to fulfill a patch test [31], the assumed stress resultants are interpolated throughout the element in the following form [26, 27]:
H
(-)in _
PaO
(n)in
o
(n)in _
0((
(n)in (W'n
((n)in " ^ 12
(27)
P.=[16 P.]
ttff 2 0 0 0 0
t12t12 f 2 t2 t2 f 0 0 0 0
0 0 f1 f 2 0 0
?? f 2 t21t-22f1 0 0 0 0
0 0 0 0 2 tlf1
0 0 0 0 t12 f 2 tUf1
(28)
where I6 is the unit matrix of order 6 x 6; ta are the elements of the Jacobian matrix J evaluated at the element center; fa are the transformed coordinates defined as:
f— 4 (<
t-\ — 1X1 X2 X3 + x
+x°); t-2a—4 (+
Xo Xn X
4 /'
1 1
; ^a—-p jj^Ad^2;
Ael -1-1
11
Ael — jjA dfMf2; A — Co + c^1 + c2^2; -1-1
C0 — 1 [(X1 - X3)(x2 - X4)- (X2 - X4)(X12 - X3)];
C1 — 1 [(xl - X2)(x3 - X4 )- (X3 - X4)(X12 - X2 )]; 8
C2 — 8[(X -x4)(X22 -X32)-(X2 -X3)(X12 -X42)]. (29)
The purpose of introducing fa lies in the simplicity of some elemental matrices of the hybrid-mixed method [32, 33] since 1 1
jjfaA dfMf2 — 0. (30)
-1-1
The displacement-independent strains are interpolated throughout the element as suggested in [26, 27]:
e( n)'n — P ^(n)'n ^(n)'n —
\J/(n)'n ^(n)'n
(n)'n 12
(31)
n
a
r
Pe =[16 Pe ],
2 4 ? 0 0 0 0
4 -4?2 2 2 <n| 0 0 0 0
0 0 f1 ?2 0 0
2 7J72 ?2 2 7? 72 f1 0 0 0 0
0 0 0 0 2 7??1
0 0 0 0 72 ?2 72?1
(32)
that corresponds to the interpolation of stress resultants (27), where ^ are the elements of the inverse Jacobian matrix evaluated at the element center:
A = ^2 , A = , ^2 = ^2, ^2 = • (33)
Substituting interpolations (18), (25), (27) and (31) in the Hu-Washizu variational equation (15), we arrive at the element equilibrium equations:
QT^(n)in _H^(«)injnkn D(n) Jn ^ (n)kn (34) Jn kn Q¥(n)in = R(n)in U
(35)
h(r(n)i„ )T0(n)in = f, (36)
n 'n
where F is the element-wise surface traction vector; Q, D(n) given by:
Q = jjPj PeA d^2,
Q, D(n)Jn and R(n)in are the elemental matrices
1 1
(37)
-1-1
11
D(n) Jn = j|PeTC(n) Jn PeA d^d?2
-1-1 11
(38)
R
(n)in -
jjPöTB(n)i'n A d?M?2.
-1-1
The use of transformed coordinates ?a in interpolations (27) and (31) is of great importance because the basic matrix of the hybrid stress-strain formulation Q becomes quasidiagonal:
Q = ^eldiag ^ «1^
"«11 «12 " A
, «22, «11
_«12 «22 _ y
where
4l = 4co;
11
(
«aß =-
5aß -
3coco y
,(39)
40)
A- ¡¡r^A d^2 = 3
Ael -1-1
Due to interpolations (27), (28), (31) and (32) the stress resultants and displacement-independent strains are discontinuous at the element boundaries.
Therefore, the column matrices 0(n)i" and ¥ (n)'n can be eliminated on the element level that yields finite element equations
KU = F, (41)
where K is the element stiffness matrix of order 12NSaSx12NSaS defined as
K = ££££A(nJ (R(n)'n )TQ-1D(n)JnQ-1R(n)kn (42)
n 'n Jn kn
Since the matrix Q is quasidiagonal, its inversion can be readily fulfilled in a closed form
Q-1 = -Uiag
Ael
(
I6, 1/«22, 1/«11,
«22/d -«12/d -«12/d «n/d
(43)
1/«22, 1/«11
where d = ronro22 - ro12ro12 . Thus, no expensive numerical inversion is needed if one uses the hybrid stress-strain quadrilateral plate formulation developed.
Remark 1. Because Ln^'n are the Lagrange polynomials of degree In -1 it is possible to carry out exact integration in (16) utilizing the Gaussian quadratures. Note also that the elemental matrices (38) are evaluated numerically through Gauss integration with 2x2 sampling points.
Remark 2. In a hybrid-mixed finite element formulation, the displacement-independent strains and stress resultants are selected so that the four-node quadrilateral plate element would be free of shear locking and kinematically stable. Owing to strain interpolation (31), we introduced 12 assumed strain parameters ^1(n)'n, ^n)'n,..., for each SaS, that
is, 12NSaS for all SaS. It seems to be excessive recalling about 3NSaS DOFs per node. However, there exist six dependent strain modes exactly, which provide a correct rank of the element stiffness matrix [26].
Benchmark Problems
The performance of the proposed hybrid-mixed four-node laminated FG plate element denoted by SaSQP4 is evaluated with exact solutions of elasticity including 3D patch tests.
Bending patch test for laminated plate. The plate patch test for the bending behavior of quadrilateral elements confirms that the finite element formulation developed is able to reproduce the constant stressstrain states for distorted mesh configurations. Here, we consider a patch of five plate elements [34] with four external and four internal nodes as depicted in Fig. 3.
e
c
c
c
c
0
0
0
0
£
Coordinate Ps P6 P7 P* a = 24, 6 = 12, ft = 0.1
X« 4 18 16 8
2 3 8 8
Fig. 3. Patch test for a laminated FG plate
To achieve the constant 3D bending stress-strain state in the case of zero Poisson's ratios, the displacements of the nth layer are chosen as:
1 2
= ex | x +—x 2
,(») = ex3 f X1
,(«) = ex3
1
= ex | - x + x
2
u\n) — -1 e 3 2
(x1 ) + x1x2 +(x2 )
(44)
Inserting (44) in strain-displacement equations (3) and using constitutive equations, we arrive at the constant strain and stress fields in surfaces parallel to the middle surface:
S(1 — 822 — ex ; 2s(2 — ex ;
4n)—0,
(45)
a(1) — a(n) —
22
— Enexa(2) — Enex3 /2; a(3n) — 0, (46)
where En is the elastic modulus of the nth layer. One can verify that the equilibrium equations, continuity conditions at interfaces and boundary conditions on outer surfaces, which are free of tractions, are satisfied exactly.
Owing to 3D solution (44), the SaS displacements at exterior nodes can be prescribed as follows:
,(n)in _ „_(n)i
1
— ezy
"nl x1 +1 x2
(n)in — ez(n)i
"nl 1 x1 + x2
v 2
(n)in
— — e 2
(x1 ) + x1x2 +(x2 )2
that results in:
(n)in _ e(n)in Sn)in-
J11
— 8
22
— ezv
2s
(n)in _ „Sn)in ■
12
— ez
,(n)in _
(47)
(48)
Ji3
— 0.
As a numerical example, we consider a sandwich plate with E = E3 = 107; E2 = 105; hx = h3 = h/10; h2 = 8h/10 and e = 10 5. Applying the prescribed displacements (47) to exterior nodes, one can observe
that the displacements and strains at interior nodes are identical to analytical answers. This is partially confirmed by displacements of the top surface u+ at interior nodes listed in Table 1 for three and five SaS inside each layer. Thus, the SaSQP4 element passes the bending plate patch test for a sandwich plate. It should be mentioned that the computations were performed employing the 16-digit calculation.
Bending patch test for FG plate. Consider again a patch of five plate elements shown in Fig. 3. The elastic modulus is assumed to be distributed through the thickness of the plate according to the exponential law
E = E ~ea( z+05);
- 0.5 < z < 0.5
(49)
where E is the elastic modulus on the bottom surface; z = x3/h is the dimensionless thickness coordinate; a is the material gradient index defined as
a = ln( + / E"), (50)
where E + is the elastic modulus on the top surface.
Applying the SaS displacements (47) for n = 1 to exterior nodes of the plate with E = 106, a = 2 and e = 10~5, one can conclude that the SaSQP4 element passes the bending patch test for a FG plate as well. It is seen from Table 2, where displacements of the top surface at interior nodes for three and seven SaS are presented.
Table 1
Displacements at interior nodes in the bending patch test for a sandwich plate
Exact values
Node 106- u+ 106- u+ -104- u3+
P5 2.500000 2.000000 1.400000
P6 9.750000 6.000000 19.35000
P7 10.00000 8.000000 22.40000
P8 6.000000 6.000000 9.600000
P5 2.500013 2.000003 1.400006
P6 9.749988 6.000002 19.35001
P7 9.999981 8.000001 22.40002
P8 6.000017 5.999995 9.600016
P5 2.499885 2.000125 1.399982
P6 9.750077 5.999927 19.34993
P7 9.999878 8.000213 22.40000
P8 6.000069 6.000015 9.600048
2
1
In = 3
In = 5
Table 2
Displacements at interior nodes in the bending patch test for a FG plate
C3333 =36 2 GPa;
C1 313 = C2323 = 10 0 GPa;
Formulation Node 106- u+ 106- u+ -104- u+
P5 2.5000000 2.0000000 1.4000000
Exact P6 9.7500000 6.0000000 19.350000
values P7 10.000000 8.0000000 22.400000
P8 6.0000000 6.0000000 9.6000000
P5 2.4999924 1.9999993 1.3999954
I1 = 3 P6 9.7500068 5.9999996 19.349998
P7 10.000003 8.0000001 22.399997
P8 5.9999974 6.0000007 9.5999966
P5 2.4999924 1.9999993 1.3999954
P6 9.7500068 5.9999982 19.349993
I1 = 7 P7 10.000010 8.0000009 22.399988
P8 5.9999903 6.0000028 9.5999871
Simply supported three-layer FG rectangular plate under sinusoidal loading. Here, we study a three-layer rectangular plate subjected to sinusoidally distributed transverse loading acting on its top surface
+ . nx1 . nx2 p3 = P0sin—sin—: a b
p3 = 0,
(51)
where a and b are the length and width of the plate.
The bottom and top layers with equal thicknesses hi = h3 = h/4, are composed of the graphite-epoxy composite with the material properties
El = 172.72 GPa, Et = 6.909 GPa, Glt = 3.45 GPa, Gtt = 1.38 GPa, and vLT = vTT = 0.25. It is assumed that the fibers of bottom and top layers are oriented respectively in x1- and x2-directions. The central layer of the thickness h2 = h/2 is made of the transversely isotropic FG material with the elastic constants distributed through the thickness according to a power law:
C,212 = 13.3 GPa,
-1212
whereas the elastic constants on the top interface are taken as j = 2j .
To analyze efficiently the results obtained for Y = 2, we introduce the dimensionless displacements and stresses at crucial points as functions of the thickness coordinate:
u1 = ELu1(a,b/2,z//hp0; u2 = ELu2(a/2,b,z//hp0;
u3 = ELu3 (a/2, b/2, z//hp0; en =cn(a/2,b/2,z//p0; c22 =a22(a/2, b/2, z//p0; ci2 =ci2(a, b, z)/ p0;
®13 =a13 (a, b/2, z / / p0; ^23 = CT23 (a/2, b, z//p0; ^33 = CT33(a/2, b/2, z//Po,
where a = b = 1m.
Due to symmetry of the problem, only one quarter of the plate (see Fig. 4) is modeled by a uniform 64x64 mesh of SaSQP4 elements. The data listed in Tables 3 and 4 show that the SaSQP4 element allows reproducing the authors' exact SaS solution [16] for thick and moderately thick plates with a high accuracy using the sufficiently large number of SaS inside the layers. As can be seen, the SaSQP4 element provides from three to four right digits for the displacements and stresses at crucial points.
Cijkl = CijklV + Ci+klV +, V + = 1 -V", V" = (0.5 -2z)Y, -0.25 < z < 0.25,
(52)
where Ci
ijkl
and C+ki are the values of elastic
constants on both interfaces; y is the material gradient index; z = x3/h is the dimensionless thickness coordinate. The elastic constants on the bottom interface are considered to be the same as those in [16]:
C-122 = 14.7 GPa;
C1133 = C2233 = 10 1 GPa;
C1111 = C2222 = 41.3 GPa;
Fig. 4. One quarter of the simply supported three-layer FG rectangular plate modeled by distorted 4kx4k meshes with ck = 8a/8k and dk = 8b/8&, where k = 1,2,..., 16 and 8e[0, 0.6]
Table 3
Convergence study for a three-layer plate with alh = 2 using a uniform 64x64 mesh
Formulation «1(0.5) «2(0.5) «3(0.5) ön(0.5) 022(0.5) 012(0.5) 013(0) 023(0) C33(0)
In = 3 3.1523 1.3399 12.772 0.48238 2.4846 -0.14093 -0.55566 -0.49464 0.43906
In = 4 3.1999 1.3734 12.832 0.47708 2.5271 -0.14347 -0.60492 -0.55892 0.44618
In = 5 3.2013 1.3769 12.836 0.47684 2.5322 -0.14363 -0.61857 -0.57944 0.45233
In = 6 3.2014 1.3769 12.836 0.47643 2.5317 -0.14363 -0.61815 -0.57703 0.45162
Exact [16] 3.2012 - 12.835 0.47646 2.5318 -0.14364 -0.61797 -0.57637 0.45156
Table 4
Convergence study for a three-layer plate with alh = 10 using a uniform 64x64 mesh
Formulation «1(0.5) «2(0.5) «3(0.5) ön(0.5) ^22 (0.5) 012(0.5) 013(0) C23(0) °33(0)
In = 3 418.71 143.34 1969.8 5.9872 46.781 -3.5266 -2.8096 -2.2369 0.48776
In = 4 418.95 143.42 1970.7 5.9834 46.798 -3.5286 -2.9218 -2.4611 0.50103
In = 5 418.95 143.42 1970.7 5.9823 46.798 -3.5286 -2.9551 -2.5383 0.50483
In = 6 418.95 143.42 1970.7 5.9823 46.797 -3.5286 -2.9635 -2.5339 0.50420
Exact [16] 418.93 - 1970.7 5.9825 46.799 -3.5288 -2.9655 -2.5329 0.50436
Figure 5 displays the distributions of stresses through the thickness of the three-layer FG plate for different values of the slenderness ratio alh by choosing five SaS and the same fine finite element mesh. These results demonstrate convincingly the high potential of the SaSQP4 element developed because
the boundary conditions on bottom and top surfaces and the continuity conditions at interfaces for transverse stresses are satisfied correctly.
To investigate the performance of the SaSQP4 element more carefully, we consider the distorted 4kx4k finite element meshes composed of 2kx2k
-2.4 -1,6_ -0.8 Stress IOai/5
Fig. 5. Through-thickness distributions of stresses of the simply supported three-layer FG square plate with I1 = I2 = I3 = 5 using a uniform 64x64 mesh, where S = alh
squares with distorted 2x2 meshes inside them as depicted in Fig. 4. The element mesh inside each square is distorted by moving the inner node along the diagonal. As a result, the generated meshes are defined
by distortion parameters ck = 8a/8k and dk = 8b/8k, which are dependent on a single distortion parameter 5e[0, 0.6]. Figures 6 a, b show the results of the convergence study due to mesh refinement and mesh
0.995
0.990
1 0.985
0.980
0.975
0.2^
oil
- LO.4
_[0.6 Ref. solution (-);
cp = -3.5288
1 ! 1 1 1 1
4 6 8 10 12 Mesh parameter k
14 16
Ref. solution (-):
e
13 "
_L_
_L
0.98
0.96
0.94
0.92
0>
e oT"^
~ I0-2
-10.4
Ref. solution (-):
i = 0.50436 ......
4 6 8 10 12 Mesh parameter k
a)
14 16
Mesh parameter k
10 12 14 16
4 6 8 10 12 14 16 Mesh parameter k
b)
Fig. 6. Convergence study due to mesh refinement and mesh distortion for a simply supported three-layer FG square plate with:
a - a/h = 10 and ^ = I2 = I3 = 5; b - a/h = 50 and ^ = I2 = I3 = 5
distortion through the use of normalized displacements and stresses for slenderness ratios of 10 and 50 by choosing five SaS inside each layer. The analytical answers are provided by the exact SaS solution [16]. It is seen that the SaSQP4 element behaves practically insensitive with respect to the extremely high mesh distortion including 5 = 0.6 except for transverse stresses in the case of thin plates.
Conclusion
The paper describes a hybrid-mixed ANS four-node quadrilateral laminated FG plate element based on the SaS formulation in which the displacements of SaS are utilized as basic plate unknowns. The SaS are located at Chebyshev polynomial nodes inside the plate body that make it possible to minimize uniformly the error due to Lagrange interpolation of displacements, strains and material properties through the layer thicknesses. The element stiffness matrix is evaluated without using the expensive numerical matrix inversion that is impossible in available hybrid-mixed finite element formulations. The quadrilateral element developed passes 3D bending patch tests for laminated and FG plates. It can be recommended for the 3D stress analysis of thin and thick laminated FG plates due to the fact that the SaS solutions asymptotically approach the solutions of elasticity as the number of SaS goes to infinity.
Acknowledgements
This work was supported by the Russian Science Foundation (Grant No. 15-19-30002) and the Russian Ministry of Education and Science (Grant No. 9.4914.2017).
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