Accepted Manuscript
Buckling and vibration of the two-dimensional quasicrystal cylindrical shells under axial compression Y.S. Li , W.J. Feng , Ch. Zhang PII: DOI: Reference:
S0307-904X(17)30357-8 10.1016/j.apm.2017.05.030 APM 11782
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
22 August 2016 9 April 2017 18 May 2017
Please cite this article as: Y.S. Li , W.J. Feng , Ch. Zhang , Buckling and vibration of the twodimensional quasicrystal cylindrical shells under axial compression, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.05.030
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ACCEPTED MANUSCRIPT
Highlights Buckling and free vibration model of 2D DC shell under axial compression is proposed.
• •
Numerical results for simply supported cylindrical DC shells are presented. Effects of various parameters of QC shell on buckling load and frequency are revealed.
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Resubmitted to the Appl. Math. Modell. in Apr. 2017(Ms. Ref. No.: APM-D-16-02092R1)
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Buckling and vibration of the two-dimensional quasicrystal cylindrical shells under axial compression Y.S. Li a, W. J. Feng b, c , Ch. Zhangc a
College of Engineering, Hebei University of Engineering, Handan 056038, PR China
b
Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043,
PR China
Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany
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c
Abstract
In this paper, the buckling and the free vibration of the quasicrystal cylindrical shells under axial compression are investigated. Three quasi-periodicity cases of
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quasicrystal cylindrical shells are considered. The first-order shear displacement theory of the cylindrical shells is utilized to obtain the equations of motion and the
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boundary conditions. Numerical results for simply supported cylindrical shells at the two ends are calculated. The effects of the geometry, in-plane phonon and phason
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loads, and half-wave number of the quasicrystal cylindrical shells on both the
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buckling loads and the frequency are demonstrated.
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Keyword: buckling; free vibration; axial compression; cylindrical shell; quasicrystal
Corresponding author. Tel: +86-311-87936541; fax: +86-311-87936466 E-mail address:
[email protected] (W.J. Feng). 2
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1. Introduction Since the quasicrystal (QC) structure with a long-range order but no periodicity was first observed by Shechtman [1], great progress has been made in the study of the elastic properties of quasicrystals (QCs) [2-10]. The physical properties, such as the structural, electronic, magnetic, optical and
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thermal properties, of quasicrystals have been investigated intensively. Elasticity is one of the interesting properties of quasicrystals. The quasicrystals include one-dimensional, two-dimensional and three-dimensional structures. For the two-dimensional quasicrystals, they are in five-dimensional space. Among many interesting topics related to QCs, the fracture properties of QC solids attracted more
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and more researchers’ interest [11-18]. In addition, numerous studies on elasticity theory of two-dimension (2D) QCs have been carried out. Among them, Gao and Zhao [19] derived the general solutions of three-dimensional (3D) problems of 2D hexagonal QCs. Li [20] formulated new explicit expressions for a general solution of
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elastohydrodynamic problems of decagonal QCs. Gao and Ricoeur [21] investigated the 3D problems associated with a spheroidal QC inclusion embedded inside an
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infinite dissimilar QC matrix subject to uniform loadings at infinity. Li et al. [22] dealt with the contact problem for a half-space of a 2D hexagonal QC punched by
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three common indenters (cylindrical flat-ended, conical and spherical punches). Reviews on the linear elasticity theory of QCs can be found in Fan [23,24].
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On the other hand, by utilizing the powerful pseudo-Stroh formalism, Yang et al.
[25,26] derived the exact closed-form solutions for simply supported and multilayered
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one-dimension (1D) and 2D decagonal QC plate under surface loadings. Furthermore, the 1D orthorhombic QC plate [27] and shallow shell [28] under the static and transient dynamic loads were, respectively, investigated by applying the meshless local Petrov-Galerkin (MLPG) method. However, to the best of authors’ knowledge, the buckling and free vibration problems of QC cylindrical shells have been not yet addressed in literature. This paper investigates the buckling and the free vibration properties of a 2D 3
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hexagonal QC cylindrical shell under axial compression. The main objective is to find the relationship between the vibration frequency and the axial force which can be used to measure the critical force from frequency by the dynamic method. The organization of this paper is as follows: In Section 2, we present the constitutive equations of QCs, and then the generalized displacement and stress fields of the QC cylindrical shell are presented in Section 3. In Section 4, we derive the equations of
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motion and the boundary conditions of the QC cylindrical shell by using the Hamilton’s principle. In Section 5, we present and discuss the numerical results, and the conclusions are drawn in Section 6.
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2. Constitutive equations of the 2-dimensional quasicrystals
According to the description of a 2-dimensional quasicrystal as a quasi-periodic structure which is in 5-dimensional space, the 5-dimensional space can be divided into two orthogonal subspaces, one being 3-dimensional physical or parallel space and the other being 2-dimensional perpendicular or complementary space. Therefore, for
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each 2-dimensional quasicrystal, in addition to the usual phonon displacements ui and
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phonon strains εij describing the local shifts of atoms in the physical space, one must introduce the phason displacements wα and phason strains wαl to describe the local
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order of atoms in the perpendicular space. For the 2D QCs with x1 and x2 as the quasi-periodic directions and x3 as the
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periodic direction referring to a rectangular Cartesian coordinate system (x1, x2, x3), the Penrose tiling of the decagonal 2D QCs in the quasi-periodic plane can be seen in
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Fig. 1. The constitutive equations of the 2D QCs relating the phonon stresses ζij, phason stresses Hβj, phonon strains εkl, and phason strains wαl are given by Fan [24] ij Cijkl kl Rijl wl ,
(1)
H j Rkl j kl K j l w l ,
(2)
where εkl denotes the phonon strain components ε11, ε22, ε33, γ23, γ31, and γ12; ζij denotes the phonon stress components ζ11, ζ22, ζ33, ζ23, ζ31, and ζ12. wαl denotes the phason strain components w11, w22, w23, w12, w13, and w21; Hβj denotes the phason 4
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stress componenets H11, H22, H23, H12, H13, and H21. In Eqs. (1) and (2), Cijkl and Kβjαl are, respectively, the elastic constants of the phonon and phason fields, and Rijαl are the coupling constants between the phonon and phason fields. For 2D decagonal QCs with the point groups 10mm, the three constant tensors can be written as 0 0 0 0
0 0
0 0
C44 0
R1 R 1 0 R 0 0 0 K1 K 2 0 K 0 0 0
0 0
R1
0 0
R1 0 0 0 0
0 0 0 0 0
, 0 C66 0 0 0 0
(3)
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C13 0 C13 0 C33 0 0 C44
0 0 0 0 0 0 , 0 0 0 0 0 R1
0 0 0 0 R1
K2
0
0
0
K1 0 0 0 0
0 K4 0 0 0
0 0 K1 0 K2
0 0 0 K4 0
(4)
0 0 . K2 0 K1 0
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C12 C11 C13 0
(5)
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C11 C 12 C C 13 0 0 0
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3. Generalized displacement and stress fields of the quasicrystal cylindrical shell
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The QC cylindrical shell considered in this paper under the axial compression with the length l, radius r, and thickness h is shown in Fig. 2. The coordinate system
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is introduced such that the x and θ coordinates are along the axial and circumferential directions, and the z axis is normal to the natural surface and pointing outward. The
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origin of the coordinate system is located at the left end in the neutral surface of the circular cylindrical shell. According to the first-order shear displacement shell theory [29], the
displacements of the phonon field for the 2D QC cylindrical shell can be written as u1 x, , z, t ux x, , t zx x, , t ,
(6)
u2 x, , z, t u x, , t z x, , t ,
(7)
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u3 x, , z, t uz x, , t .
(8)
Three cases of the 2D quasicrystal shells are considered in this paper. For case 1, the quasi-periodic directions are x and θ, and x-and z-directions are quasi-periodic for case 2 and θ and z for case 3. If the periodic direction of the 2D QC is in the z-direction or the thickness direction of the cylindrical shell (Case 1), the
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displacements of the phason field can be expressed as w1 x, , z, t wx x, , t z x x, , t ,
(9)
w2 x, , z, t w x, , t z x, , t .
(10)
If the periodic direction of the 2D QC is in the θ-direction of the cylindrical shell
w1 x, , z, t wx x, , t z x x, , t , w3 x, , z, t wz x, , t .
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(Case 2), the displacements of the phason field are expressed as
(11) (12)
In Case 3, the periodic direction of the 2D QC is in the x-direction of the cylindrical
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shell, and the displacements of the phason field are given by
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w2 x, , z, t w x, , t z x, , t . w3 x, , z, t wz x, , t .
(13) (14)
ux z x , x x
(15)
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xx
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The linear phonon strains for the QC cylindrical shell are given by [30]
1 u uz z , r
(16) (17)
uz , x
(18)
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1 uz u , r r
z
zx x
x
1 ux u z x z . r x r x
(19)
The phason strains of the QC cylindrical shell for the three cases mentioned above can be expressed as
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Case 1: wx x z , x x
wxx
x 1 w wx x z r
(20) ,
(21)
wxz x ,
(22)
w z , x x
1 w w z r
(23)
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w x
,
(24)
w z ;
(25)
Case 2:
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wx x z , x x
wxx
x 1 w wx x z , r
wz , x
wz
1 wz , r
Case 3:
w z , x x
,
(33)
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1 w w z r
(29)
(32)
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w x
(28)
(31)
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wzz 0;
(27)
(30)
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wzx
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wxz x ,
(26)
w z ,
(34)
wzx
wz , x
(35)
wz
1 wz , r
(36)
wzz 0.
(37)
Correspondingly, the generalized stresses for the plane elasticity of the QC for
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the three cases are, respectively, obtained from Eqs. (1) and (2) as Case 1: ux z x x x
C12 u r uz z
x wx R1 x z x
R1 w r z
,
(38)
ux z x x x
C11 u r uz z
x wx R1 x z x
R1 w r z
,
(39)
xx C11
C12
zx C44 x
uz x
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1 uz u , r r
(40)
,
(41)
1 ux u z x z x r x r
x R1 wx r z
x C66
w R1 x z x
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z C44
,
(42)
R u x K 2 w u w H xx R1 x z x 1 uz z K1 x z z , x r x r x x
(43)
1 ux u z x H x R1 z r x r x
(44)
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x K1 wx r z
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H xz K4 x ,
x K 2 wx r z
,
(45) w K1 x z x
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1 ux u z x H x R1 z x r x r
w K 2 x z x
,
(46)
(47)
H z K4 ;
(48)
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R u x K1 w u w H R1 x z x 1 uz z K 2 x z z , x r x r x x
Case 2:
ux z x x x
xx C11
C13 u r uz z
x wx R1 x z x
C u ux z x 33 uz z , x r x
C13
z C44
,
(49)
(50)
1 uz u , r r
(51) 8
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uz x
zx C66 x
wz R1 x R1 x ,
(52)
1 ux u z x z x r x r
,
x C66
u H xx R1 x z x x x
(54)
u H zx R1 x z x
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x K 4 wx z , r
u H xz R1 x z x
H z
,
(55)
wz K1 x K 2 x ,
(56)
wz K 2 x K1 x ,
K 4 wz , r
x wx K 2 x z x
;
(57) (58) (59)
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u H zz R1 x z x x x
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H x
x wx K1 x z x
(53)
Case 3:
C13 u r uz z
,
ux z x x x
C11 u r uz z
R1 w r z
(60)
z C66
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zx C44 x
,
(61)
R w 1 uz u R1 1 z , r r r
(62)
uz x
(63)
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C13
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ux z x x x
xx C33
,
1 ux u z x z x r x r
x C44
,
(64)
w H x K 4 z , x x
(65)
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H
R1 u uz z r
K1 w r z
,
(66)
K w 1 uz u H z R1 K1 2 z , r r r H zx K 4
(67)
wz , x
(68)
K w 1 uz u H z R1 K 2 1 z , r x r R1 u uz z r
K 2 w r z
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H zz
(69)
.
(70)
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4. Equations of motion and boundary conditions of the QC cylindrical shell
In order to obtain the equations of motion and the boundary conditions, the Hamilton’s principle
T U W dt 0 t2
t1
(71)
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is used, where the virtual strain energy δU, the virtual work δW done by the external forces, and the virtual kinetic energy δT can be easily obtained.
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The virtual strain energies of the QC cylindrical shell for the considered three
Case 1:
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cases are, respectively,
u x N x u x u N u Q z N xx x r N x x r r u x M x x u z Q z u z N Qzx u z M xx Qzx x x r r x r M wx Yx wx w Y w M x Q z Yxx Y x x r x r x r x Tx x T Txx S xz x T x S z rdxd ; x r x r
U
2
L
0
AC
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0
Case 2:
10
(72)
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u x N x u x u N u Q z N xx x r N x x r r u x M x x u z Q z u z N Qzx u z M xx Qzx x x r r x r M wx Yx wx M x Q z Yxx x r x r x Tx x wz S z wz Txx S xz x S zx rdxd ; x r x r
U
2
0
L
0
(73)
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Case 3: u x N x u x u N u Q z N xx x r N x x r r u x M x x u z Q z u z N Qzx u z M xx Qzx x x r r x r M w Y w M x Q z Y x x r x r T wz S z wz T x S z S zx rdxd . x r x r 2
0
L
0
In Eqs. (72)-(74),
N xx , N , N x h 2 xx , , x dz, h2
M xx , M , M x h 2 xx , , x zdz, Q z , Qzx h 2 z , zx dz,
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h2
M
h2
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U
(74)
(75) (76) (77)
Yxx , Yx , Y x , Y , Yzz h 2 H xx , H x , H x , H , H zz dz,
(78)
Txx , Tx , T x , T h 2 H xx , H x , H x , H zdz,
(79)
Sxz , S z , Szx , Sz h 2 H xz , H z , H zx , H z dz,
(80)
h2
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h2
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h2
where κ is the shear correction factor.
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The virtual kinetic energies of the QC cylindrical shell for the three cases can be,
respectively, given as Case1: T
2 2 2 u x x u u z 1 z z 2 t t t t t
2 2 x w wx z z d ; t t t t
Case 2: 11
(81)
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T
2 2 2 u x x u u z 1 z z 2 t t t t t
(82)
2 2 x wz wx z d ; t t t
Case 3: T
2 2 2 u x x u u z 1 z z 2 t t t t t
(83)
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2 2 wz w z d . t t t
The virtual work δW carried out by the external loads acting on the QC cylindrical shell for Case1 is L
0
2
0
N 1xx
uz uz rdxd, x x
(84)
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W
whilst for Cases 2 and 3, W
L
0
2
0
1 uz uz 2 wz wz N xx x x N xx x x rdxd ,
(85)
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where N 1xx and N xx2 denote the in-plane forces.
Now, employing the Hamilton’s principle the governing equations of motion of N xx 1 N x 2u I 0 2x , x r t
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the 2D QC cylindrical shell can be formulated as (86) (87)
Qzx 1 Q z N 2u 2u N 1xx 2z I 0 2z , x r r x t
(88)
M xx 1 M x 2 Qzx I 2 2x , x r t
(89)
M x 1 M 2 Q z I 2 2 . x r t
(90)
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CE
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N x 1 N Q z 2u I 0 2 , x r r t
And for the three different cases, the governing equations of motion also, respectively, include Case 1: Yxx 1 Yx 2 w I0 2 x , x r t
(91)
12
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(92)
Txx 1 Tx 2 S xz I 2 2 x , x r t
(93)
T x 1 T 2 S z I 2 2 ; x r t
(94)
Case 2: Yxx 1 Yx 2 w I0 2 x , x r t
(95)
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S zx 1 S z 2 wz 2 wz N xx2 I , 0 x r x 2 t 2
(96)
Txx 1 Tx 2 S xz I 2 2 x ; x r t
(97)
Case 3:
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Y x 1 Y 2 w I 0 2 , x r t S zx 1 S z 2 wz 2 wz N xx2 I , 0 x r x 2 t 2 T x 1 T 2 S z I 2 2 . x r t
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In Eqs. (86)-(100),
I 0 , I 2 h 2 1, z 2 dz. h2
(98) (99)
(100)
(101)
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Substituting Eqs. (75)-(79) into Eqs. (86)-(100) and using Eqs. (38)-(70), one further obtains the equations of motion of the QC cylindrical shell in terms of the
Case 1:
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2 u x A66 2 u x A12 A66 2 u A u 2 12 z r x r x x 2 r 2 2 2 2 wx D1 wx 2 D1 w 2ux D1 I , 0 r x x 2 r 2 2 t 2
(102)
AC
A11
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displacement variables as
A12 A66
2ux 2 u A11 2 u A44 A A uz A44 A66 2 2 u 11 2 44 2 2 r x r x r r r 2 w D1 2 w 2 u 2 D 2 wx 1 D1 I , 0 r x x 2 r 2 2 t 2
A12 u x A11 A44 u 2u z A44 2u z A44 2 2 r x r x 2 r 2 A44 D1 wx D1 w A 2uz 2u 112 u z A44 x 2 N 1xx I 0 2z , 2 x r r x r r x t
13
(103)
(104)
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A44
(105)
2 A44 uz B12 B66 2 x B66 r r r x x 2 2 2 2 E1 2 2 B 2 E1 x 11 A E I , 44 1 2 r x r 2 2 x 2 r 2 2 t 2
A44
u
(106)
(107)
2u D 2u D1 uz 2 w F1 2 w 2 w 2 D1 2ux D1 2 21 F I , 1 0 r x x r 2 r 2 x 2 r 2 2 t 2
(108)
2 x E1 2 x 2E1 2 2 G 2 x 2 2 G1 2x 21 F4 x I 2 2 x , 2 2 2 r x x r x r t
(109)
2 x 2 E 2 2 G1 2 2 E1 2 21 G F I ; 1 4 2 x x r 2 x 2 r 2 2 t 2
(110)
2 E1
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E1
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2ux D1 2ux 2 D1 2u D1 uz 2 wx F1 2 wx 2 wx F I , 1 0 r x r x x 2 r 2 2 x 2 r 2 2 t 2
D1
Case 2:
2ux A44 2ux A13 A44 2u A13 uz 2 wx 2ux D I , 1 0 r x r x x 2 r 2 2 x 2 t 2 A13 A44 2ux 2u A 2u A44 A33 A44 uz A44 2u A44 2 332 u I , 0 r x r x r 2 r2 r2 t 2 A11
M
A13 u x A33 A44 u 2 u z A44 2u z A 2 66 r x r2 x 2 r 2 2 A x A44 x 2 wz 2uz 1 uz 33 u A D D N I , z 66 1 1 xx 0 x r x r2 x 2 x 2 t 2
ED
2 x B66 2 x u z B11 2 A66 x x x 2 r 2 B B44 2 2 x 2 x w 13 D1 z E1 D1 x I 2 , 2 r x x x t 2
PT
A66
AC
CE
2 B 2 2 A44 A u B B44 2 x u 44 z 13 B44 2 332 A I , 44 2 r r r x x r 2 t 2 2u 2 w F 2 wx 2 wx D1 2x F1 2x 24 I , 0 x x r 2 t 2 x 2u 2 w F 2 wz 2 wz 2 w D1 2z D1 x F1 2z 2 4 F2 N xx2 I0 2 z , 2 2 x x x x r x t 2 2 2 2 x u w G x D1 z E1 2x D1 x F2 z G1 2x 24 F I ; 1 x 2 x x x x r 2 t 2
(111) (112)
(113)
(114)
(115) (116) (117) (118)
Case3: A33
2ux A44 2ux A13 A44 2u A13 uz 2ux I , 0 r x r x x 2 r 2 2 t 2
14
(119)
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2ux 2 u A11 2 u A66 A44 2 2 u r x x 2 r 2 r 2 A A uz A66 2 u D w D1 wz D1 11 2 66 21 I , 0 r r r r 2 r 2 t 2
A13 u x A13 A66 u 2 u z A66 2 u z A 2 44 r x r2 x 2 r 2 2 x A66 D1 w D1 2 wz D1 A 2uz 1 uz 11 u A N I , z 44 xx 0 x r r r2 r 2 r 2 2 x 2 t 2
2 B 2 x B13 B44 2 2 x uz B33 2x 442 A I , 44 x 2 x r x x r 2 t 2
A44
A66 uz B13 B44 2 x 2 B44 r r r x x 2 2 2 2 B D1 wz E1 11 A D I , 66 1 2 r r 2 2 r 2 2 t 2
A66
u
2 w F1 2 w 2 w D1 2u D1 uz F I , 4 0 r 2 2 r 2 x 2 r 2 2 t 2
(121)
(122)
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(120)
(123)
(124)
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2 D1 u D1 2uz D1 2 wz F1 2 wz F2 2 wz 2 wz F N I , (125) 4 xx 0 r r r 2 r 2 2 x 2 r 2 2 x 2 t 2 2 G1 2 2 D D u E 2 F2 wz 1 u 1 z 21 D G K h I . (126) 1 4 1 2 r r r 2 r x 2 r 2 2 t 2
A , D , F C h, R h, K h, ij
i
i
ij
i
i
Cij h3 Ri h3 Ki h3 , , . 12 12 12
ij
i
i
ED
B , E , G
M
In the above equations, the following quantities are introduced
(127) (128)
obtained as
PT
Also, the boundary conditions corresponding to the equations of motion are
N x n 0, r
(129)
N n 0, r
(130)
CE
ux 0 or N xx nx
u 0 or N x nx
AC
uz 0 or Qzx nx
Q z n 0, r
(131)
x 0 or M xx nx
M x n 0, r
(132)
0 or M x nx
M n 0, r
(133)
where nx and nθ denote the direction cosines of the unit normal to the boundary of the middle plane. In addition, for Case 1, we have 15
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wx 0 or Yxx nx
Yx n 0, r
(134)
w 0 or Yx nx
Y n 0, r
(135)
x 0 or Txx nx
Tx n 0, r
(136)
0 or Tx nx
T n 0. r
(137)
wx 0 or Yxx nx
Yx n 0, r
x 0 or Txx nx
Tx n 0, r
wz 0 or S zx nx
Tz n 0. r
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For Case 2, we obtain (138) (139) (140)
Y n 0, r
0 or Tx nx
T n 0, r
wz 0 or S zx nx
Tz n 0. r
(141) (142) (143)
M
w 0 or Yx nx
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For Case 3, one has
For the considered simply supported QC cylindrical shell, the following
ED
boundary conditions are adopted
(144)
PT
N xx u uz M xx 0, Yxx w wz Txx 0.
To solve the governing dynamic equations with the aforementioned boundary
CE
conditions, the phonon and phason displacements and rotations are expressed as M
N
ux uxmn cos m x sin n exp it ,
(145)
m 1 n 1 M
N
u u mn sin m x cos n exp it ,
AC
(146)
m 1 n 1 M
N
uz uzmn sin m x sin n exp it ,
(147)
m 1 n 1 M
N
x xmn cos m x sin n exp it ,
(148)
m 1 n 1 M
N
mn sin m x cos n exp it ,
(149)
m 1 n 1 M
N
wx wxmn cos m x sin n exp it ,
(150)
m 1 n 1
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N
w w mn sin m x cos n exp it ,
(151)
m 1 n 1 M
N
wz wzmn sin m x sin n exp it ,
(152)
m 1 n 1 M
N
x xmn cos m x sin n exp it ,
(153)
m 1 n 1 M
N
mn sin m x cos n exp it ,
(154)
m 1 n 1
denotes the number of the axial half-waves.
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where m m l , n represents the number of the circumferential waves and m
Substituting Eqs. (145)-(154) into Eqs. (102)-(126), the governing dynamic equations of the QC cylindrical shell can be rewritten as
K M w 0, 2
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(155)
where the elements kij and mii of the stiffness matrix K and the diagonal mass matrix M can be found in the Appendices A, B and C. The extended displacement vector w is, respectively, defined as w ux , u , uz , x , , wx , w , x ,
for Case 1, w ux , u , uz , x , , wx , wz , x
T
(157)
ED
for Case 2, and
T
(158)
CE
PT
w ux , u , uz , x , , w , wz ,
for Case 3.
(156)
M
T
For the buckling problem of the cylindrical shell under axial compression, we
AC
should delete the term exp(-iωt) in Eqs. (145)-(154) and Eq. (155) becomes Kw 0.
(159)
The buckling load can be obtained from the equation above. 5. Numerical results In this section, some typical numerical results for the buckling and free vibration of the QC cylindrical shell will be presented. 17
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Firstly, as known, if the elements of the phason field are omitted, the governing equations of motion and the boundary conditions can be reduced to the corresponding ones for the classical elastic cylindrical shells. Therefore, to check the validation and accuracy of the present analysis, we compute the natural frequencies for an elastic cylindrical shell and compared with the results given by Alibeigloo and Shaban [31] in Table 1. An excellent agreement between these results can be observed.
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In the following calculations, the used material properties for the 2D hexagonal QC are given in Table 2 [25].
5.1 Buckling of the QC cylindrical shell under axial compression
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It is assumed that the external in-plane loads are N 1xx 1 p, N xx2 2 p . The in-plane load is normalized by P
12 p . c11h
(160)
The buckling load is denoted by Pcr. For the buckling problem of the cylindrical shell,
M
the time-dependent terms can be omitted.
The buckling loads Pcr of the 2D hexagonal QC cylindrical shell under axial
ED
compression for Case 1 are listed in Table 3. It is found that increasing the radius of the cylindrical shell can decrease the buckling loads.
PT
The buckling loads Pcr of the 2D hexagonal QC cylindrical shell under axial compression for Case 2 are presented in Table 4. It is interesting to note that there are
CE
two buckling loads at the same buckling mode, which is quite different from the case of a purely elastic shell [32]. The possible reason for this is that for the present
AC
problem, there are two in-plane stresses, phonon and phason stresses, in the axial direction of the QC cylindrical shell. Table 4 also shows that for fixed length-radius ratio l/r and radius-thickness ratio r/h, one of the two buckling loads remains unchanged for different phason-phonon load ratios γ2/γ1, which should be related to the phonon field, and the other related to the phason field decreases with increasing γ2/γ1. As in Case 1, the buckling loads for Case 2 also decrease with increasing r/h. However, for a larger value of γ2/γ1, r/h has only a little influence on one of the 18
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buckling loads. Table 5 lists the buckling loads Pcr of the 2D hexagonal QC cylindrical shell under axial compression for Case 3. Here, similar conclusions can be drawn as in Case 2. For the fixed length-thickness ratio l/h=10, Fig. 3 shows the buckling loads Pcr of the cylindrical shell with respect to the length-radius ratio l/r for case 1, 2, and 3. one
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of the buckling loads for case 2 or 3 increases monotonically with increasing length-radius ratio, but the other one for case 2 or 3 and buckling load for case 1 increase at first then drops as length-radius ratio increases.
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5.2 Vibration of the QC cylindrical shell under axial compression In this section, the vibration frequency is normalized as
h . c11
(161)
Figure 4 shows that the variation of the normalized vibration frequency Ω of the
M
QC cylindrical shell for Case 1 with respect to the in-plane load P for the length-radius ratio l/r=1 and different radius-thickness ratios r/h. It is observed that
ED
the vibration frequency decreases with increasing in-plane load and reaches the minimum value when the in-plane load approaches to the buckling load Pcr, which has
PT
been listed in Table 3. This is because that the existence of the in-plane load weakens the stiffness of the QC cylindrical shell. The normalized vibration frequencies Ω of
CE
the QC cylindrical shell for Case 1 versus the in-plane load P for l/r=2 and l/r=3 are plotted in Figs. 5 and 6. From these figures, it can be found that the normalized
AC
vibration frequencies of the QC cylindrical shell without the in-plane load are, respectively, 0.2013, 0.1063 and 0.0562 for r/h=10 and l/r=1, 2 and 3. Fig. 7 displays the variation of the normalized vibration frequency Ω of the QC cylindrical shell for Case 1 with respect to the axial half-wave number m for different in-plane loads P. Obviously, the normalized frequency increases with increasing axial half-wave number. Figures 8-10 demonstrate the normalized vibration frequencies Ω of the QC 19
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cylindrical shell for Case 2 versus the in-plane load P for different phason-phonon load ratios γ2/γ1 for l/r=1, 2 and 3, respectively. According to these figures, the normalized frequency decreases with increasing in-plane load and reaches the minimum value as the in-plane load approaches to the first buckling load (phonon buckling load), then the frequency becomes a peak value and drops gradually up to the second buckling load (phason buckling load). It is found that the peak values of
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the frequencies for l/r=1, 2 and 3 are, respectively, 0.1812, 0.0854 and 0.0638 when γ2/γ1=1, 0.1569, 0.0632 and 0.0566 if γ2/γ1=2, and 0.0903, 0.0200 and 0.0382 as γ2/γ1=4.
Figure 11 depicts the normalized vibration frequency Ω of the QC cylindrical
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shell for Case 2 versus the in-plane load P for different r/h. In this figure, there is an extreme value in each curve when P =1.25. However, the extreme value becomes smaller as the phason-phonon load ratio increases. Fig. 12 displays the normalized frequency Ω versus the in-plane load P for different axial half-wave number m. It is
M
found that the normalized frequencies without the in-plane load and the peak value increase with increasing axial half-wave number. It is also seen from Fig. 12 that the
ED
normalized frequencies of the QC cylindrical shell without the in-plane load are 0.0860, 0.1436 and 0.2459 for m=1, 2 and 3, respectively, whilst the corresponding
PT
peak values are, respectively, 0.0903, 0.2785 and 0.3171. Figures 13-15 illustrate the normalized vibration frequencies Ω of the QC
CE
cylindrical shell for Case 3 versus the in-plane load P for l/r=1, 2 and 3 respectively. The frequencies without the axial compression are independent of γ2/γ1 for a fixed l/r,
AC
and decrease with increasing length-radius ratio. They are, respectively, 0.0862, 0.0450 and 0.0197 as l/r=1, 2, 3. As shown in Fig. 13, for different γ2/γ1, the normalized frequencies at the phonon buckling load are all equal to 0.0191. Figs. 14 and 15 indicate the same phonon buckling load but three different phason bucking loads (also see Table 5). Figure 16 shows the normalized frequency Ω of the QC cylindrical shell for Case 3 versus the in-plane load P for different r/h. The frequencies of the QC cylindrical 20
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shell without the in-plane load are, respectively, 0.1837, 0.0862 and 0.0413 for r/h=5, 10 and 20. The peak values of the frequency after the phonon buckling load are 0.1475, 0.0581 and 0.0262, respectively. The normalized frequency Ω of the QC cylindrical shell for Case 3 versus the in-plane load P for different axial half-wave number m is illustrated in Fig. 17. As shown in Fig. 17, the frequency increases with increasing axial half-wave number. And the phonon buckling load decreases with
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increasing axial half-wave number. In addition, the frequencies without the in-plane load are, respectively, 0.0862, 0.1426 and 0.2042 for m=1, 2 and 3, and after the phonon buckling load they are 0.0581, 0.0960 and 0.1919, respectively.
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6. Conclusions
In this paper, the buckling and the free vibration of the quasicrystal (QC) cylindrical shell under axial compression are investigated. The first-order shear displacement theory of the cylindrical shell is utilized to obtain the equations of motion and the boundary conditions. Numerical results for simply supported
M
boundary conditions at the two ends of the QC cylindrical shell are calculated. The
ED
effects of the geometry, in-plane phonon and phason loads, and half-wave number of the QC quasicrystal shell on both the buckling loads and the frequency are analyzed. The following main conclusions can be drawn based on the numerical results.
PT
(i) When the periodic direction is in the in-plane (x or θ direction) of the 2D QC
CE
cylindrical shell, there are two buckling loads (phonon and phason buckling loads) for the same buckling mode.
AC
(ii) The frequencies of the QC cylindrical shell decrease with increasing axial
load and reach the minimum value when the in-plane load approaches to the buckling load.
(iii) If the in-plane direction is the periodic direction, the vibration frequency shows two minima at the phonon and phason buckling loads. Acknowledgments This work is supported by the Scientific Research Fund of Hebei Education 21
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Department (ZD2014033) and the Natural Science Foundation of China (11572358, 10772123). Appendix A C66 h 2 C C66 h C h n , k12 12 m n, k13 122 m2 , k14 k15 0, 2 r r r R1h 2 2 R1h 2 k16 R1hm 2 n , k17 m n, k18 k19 0, r r k11 C11hm2
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C C h C11h 2 C44 h C44 h n , k23 11 2 44 n, k24 0, k25 , r r2 r2 r 2R h Rh k26 1 m n, k27 R1hm2 12 n 2 , k28 k29 0, r r k22 C66 hm2
C44 h
C11h N 1xx m2 , k34 C44 hm , r r2 C h Rh Rh k35 44 n, k36 1 m , k37 12 n, k38 k39 0, r r r 2
n2
(A.2)
(A.3)
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k33 C44 hm2
(A.1)
C C66 h C11h3 2 C66 h3 2 m n C44 h, k45 12 m n, 2 12 12r 12r R h3 R h3 R h3 k46 k47 0, k48 1 m2 1 2 n 2 , k49 1 m n, 12 6r 12r
(A.4)
C66 h3 2 C11h3 2 m n C44 h, k56 k57 0, 12 12r 2 R h3 R h3 R h3 k58 1 m n, k59 1 m2 1 2 n 2 , 6r 12 12r
(A.5)
3
k44
ED
M
k55
K1h 2 n , k67 k68 k69 0, r2
(A.6)
k77 K1hm2
K1h 2 n , k78 k79 0, r2
(A.7)
K1h3 2 K1h3 2 m n K 4 h, k89 0, 12 12r 2
(A.8)
K1h3 2 K1h3 2 m n K 4 h. 12 12r 2
(A.9)
CE
k88
PT
k66 K1hm2
k99
AC
m11 m22 m33 m66 m77 h, m44 m55 m88 m99
h3 12
.
(A.10)
Appendix B
C C44 h C h C44 h 2 n , k12 13 m n, k13 132 m , 2 r r r k14 k15 0, k16 R1hm2 , k17 k18 0, k11 C11hm2
22
(B.1)
ACCEPTED MANUSCRIPT C33 h 2 C44 h C C h n , k23 33 2 44 n, r2 r2 r C44 h k24 0, k25 , k26 k27 k28 0, r k22 C44 hm2
k33 C66 hm2 k35
C44 h r
C44 h r
2
n2
(B.2)
C33 h , k34 C66 hm , r2
(B.3)
n, k36 0, k37 R1hm2 , k38 R1hm ,
C C44 h C11h3 2 C66 h3 2 m n C66 h, k45 13 m n, 12 12r 12r 2 R h3 k46 0, k47 R1hm , k48 R1h 1 m2 , 12 3
C44 h3 2 C33 h3 2 m n C44 h, k56 k57 k58 0, 12 12r 2
K4 h 2 n , k67 k68 0, r2
k66 K1hm2
k77 K1hm2 k88
(B.4)
(B.5) (B.6)
K4 h 2 n , k78 K 2 hm , r2
K1h3 2 K 4 h3 2 m n K1h. 12 12r 2
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k55
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k44
h3 12
.
(B.8) (B.9)
M
m11 m22 m33 m66 m77 h, m44 m55 m88
(B.7)
Appendix C
C C44 h C44 h 2 n , k12 13 m n, 2 r r
ED
k11 C33 hm2
(C.1)
PT
C h k13 132 m , k14 k15 k16 k17 k18 0, r
C C h C11h 2 C66 h n , k23 11 2 66 n, k24 0, 2 2 r r r C66 h R1h 2 R1h R h k25 , k26 2 n , k27 2 n, k28 1 , r r r r
CE
k22 C44 hm2
C h C11h , k34 C44 hm , k35 66 n, 2 r r r R1h R1h 2 R1h k36 2 n, k37 2 n , k38 n, r r r
C66 h 2
n2
AC
k33 C44 hm2
(C.2)
(C.3)
k44
C33 h3 2 C44 h3 2 C C44 h m n C44 h, k45 13 m n, k46 k47 k48 0, 2 12 12r 12r
(C.4)
k55
C44 h3 2 C11h3 2 R1h R1h3 2 m n C h , k 0, k n , k n R1h, 66 56 57 58 12 r 12r 2 12r 2
(C.5)
k66 K 4 hm2
3
K1h 2 n , k67 k68 0, r2
(C.6)
23
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k77 K 4 hm2 k88
K1h 2 n , k78 0, r2
(C.7)
K 4 h3 2 K1h3 2 m n K1h. 12 12r 2
(C.8)
m11 m22 m33 m66 m77 h, m44 m55 m88
h3 12
(C.9)
.
References
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[1] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951-1953. [2] P.
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[3] P.
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Elasticity and dislocations in pentagonal and icosahedral quasicrystals, Phys. Rev.
ED
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PT
[6] F. Gahler, J. Rhyner, Equivalence of the generalized grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A 19 (1986) 267-277.
CE
[7] J.E.S. Socolar, Simple octagonal and dodecagonal quasicrystals, Phys. Rev. B 39 (1989) 10519-10551.
AC
[8] D.H. Ding, W.G. Yang, C.Z. Hu, R.H. Wang, Generalized elasticity theory of quasicrystals, Phys. Rev. B 48 (1993) 7003-7010.
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Philos. Mag. A 79 (1999) 1943-1952. [12] W.M. Zhou, T.Y. Fan, Plane elasticity problem of two-dimensional octagonal quasicrystals and crack problem, Chin. Phys. 10 (2001) 743-747. [13] E. Radi, P.M. Mariano, Stationary straight cracks in quasicrystals, Int. J. Fract. 166 (2010) 105-120. [14] W. Li, T.Y. Fan, Exact solutions of the generalized Dugdale model of
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two-dimensional decagonal quasicrystals, Appl. Math. Comput. 218 (2011) 3068-3071.
[15] J.H. Guo, Z.X. Lu, Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals, Appl. Math. Comput. 217
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(2011) 9397-9403.
[16] J.H. Guo, J. Yu, Y.M. Xing, Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip, Mech. Res. Comm. 52 (2013) 40-45.
M
[17] G.E. Tupholme, Row of shear cracks moving in one-dimensional hexagonal quasicrystalline materials, Eng. Fract. Mech. 134 (2015) 451-458.
ED
[18] Y.W. Wang, T.H. Wu, X.Y. Li, G.Z. Kang, Fundamental elastic field in an infinite medium of two-dimensional hexagonal quasicrystal with a planar crack:
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3D exact analysis, Int. J. Solids Struct. 66 (2015) 171-183. [19] Y. Gao, B.S. Zhao, General solutions of three-dimensional problems for
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two-dimensional quasicrystals, Appl. Math. Model. 33 (2009) 3382-3391. [20] X.F. Li, A general solution of elasto-hydrodynamics of two-dimensional
AC
quasicrystals, Phil. Mag. Lett. 91 (2011) 313-320.
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[24] T.Y. Fan, Mathematical Theory and Methods of Mechanics of Quasicrystalline Materials, Engineering 5 (2013) 407-448. [25] L.Z. Yang, Y. Gao, E. Pan, N. Waksmanski, An exact solution for a multilayered two-dimensional decagonal quasicrystal plate, Int. J. Solids Struct. 51 (2014) 1737-1749. [26] L.Z. Yang, Y. Gao, E. Pan, N. Waksmanski, An exact closed-form solution for a
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[28] J. Sladek, V. Sladek, Ch. Zhang, M. Wünsche, Modelling of orthorhombic quasicrystal shallow shells, Eur. J. Mech. A/Solids 49 (2015) 518-530. [29]G.G. Sheng, X. Wang, An analytical study of the non-linear vibrations of functionally graded cylindrical shells subjected to thermal and axial loads,
M
Compos. Struct. 97 (2013) 261–268.
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ED
shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory, Compos. Struct. 120 (2015) 65–78.
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[31] A. Alibeigloo, M. Shaban, Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity, Acta Mech. 224 (2013) 1415-1427.
AC
CE
[32] A.W. Leissa, Vibration of Shells, Washington, DC: NASA SP-288, 1973.
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Table 1 Comparisons of the natural frequency for a cylindrical shell r E (l/r=1, n=1)
1
Alibeigloo and Shaban (2013) 0.913
2
0.762
0.766
3
0.699
0.701
m
0.1
Present results
h/r
1
Alibeigloo and Shaban (2013) 0.993
2
0.936
0.937
3
0.999
1.003
m
0.916 0.2
Present results 0.995
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h/r
Table 2
Material constants for a special 2D hexagonal QC (Yang et al., 2014) (Cij, Ri and Ki in
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109Nm-2, ρ in 103kgm-3) C11
C12
C13
C33
C44
234.33
57.41
66.63
232.22
70.19
K1
K2
K4
ρ
0.8846
122
24
12
5.08
M
Table 3
R1
Buckling loads Pcr of the 2D hexagonal QC cylindrical shell under axial compression
l/r
5 1.3508 2.0957 1.7032
r/h
10
20
30
40
50
1.0548 1.9264 1.4526
0.9689 1.8630 1.3347
0.9518 1.8452 1.2965
0.9456 1.8370 1.2776
0.9427 1.8322 1.2664
AC
CE
PT
1 2 3
ED
for Case 1 (m=n=1)
Table 4 Buckling loads Pcr of the 2D hexagonal QC cylindrical shell under axial compression for Case 2 (m=n=1) l/r
γ2/γ1
r/h 5
10
15
20 27
25
30
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2 4 1
2
2 4 1
3
2 4
0.9936 4.9925 0.9936 2.4963 0.9936 1.2481 1.5979 5.1447 1.5979 2.5723 1.5979 1.2862 0.9008 5.4005 0.9008 2.7002 0.9008 1.3501
Table 5
0.9307 4.9914 0.9307 2.4957 0.9307 1.2479 1.5494 5.1444 1.5494 2.5722 1.5494 1.2861 0.8078 5.4004 0.8078 2.7002 0.8078 1.3501
0.9077 4.9911 0.9077 2.4955 0.9077 1.2478 1.5273 5.1443 1.5273 2.5727 1.5273 1.2867 0.7618 5.4003 0.7618 2.7002 0.7618 1.3501
0.8967 4.9909 0.8967 2.4954 0.8967 1.2477 1.5148 5.1443 1.5148 2.5721 1.5148 1.2861 0.7344 5.4003 0.7344 2.7001 0.7344 1.3501
0.8906 4.9908 0.8906 2.4954 0.8906 1.2477 1.5068 5.1443 1.5068 2.5721 1.5068 1.2861 0.7162 5.4003 0.7162 2.7001 0.7162 1.3501
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1
1.2947 4.9983 1.2947 2.4992 1.2495 1.2947 1.7778 5.1462 1.7778 2.5731 1.7778 1.2867 1.1877 5.4012 1.1877 2.7006 1.1877 1.3503
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1
M
Buckling loads Pcr of the 2D hexagonal QC cylindrical shell under axial compression
γ2/γ1 1 2
1.2884 1.0260 1.2884 0.5130 1.2884 0.2565 1.7766 2.5878 1.7766 1.2939 1.7766 0.6469 1.1736 5.1908 1.1736 2.5954
CE
1
5
4
AC
1
2
2 4 1
3 2
r/h
10
15
20
25
30
0.9977 1.0260 0.9977 0.5130 0.9977 0.2565 1.5971 2.5878 1.5971 1.2939 1.5971 0.6469 0.8850 5.1908 0.8850 2.5954
0.9358 1.0260 0.9358 0.5130 0.9358 0.2565 1.5483 2.5878 1.5483 1.2939 1.5483 0.6469 0.7912 5.1908 0.7912 2.5954
0.9130 1.0260 0.9130 0.5130 0.9130 0.2565 1.5261 2.5878 1.5261 1.2939 1.5261 0.6469 0.7449 5.1908 0.7449 2.5954
0.9021 1.0260 0.9021 0.5130 0.9021 0.2565 1.5135 2.5878 1.5135 1.2939 1.5135 0.6469 0.7172 5.1908 0.7172 2.5954
0.8961 1.0260 0.8961 0.5130 0.8961 0.2565 1.5054 2.5878 1.5054 1.2939 1.5054 0.6469 0.6988 5.1908 0.6988 2.5954
PT
l/r
ED
for Case 3 (m=n=1)
28
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1.1736 1.2977
0.8850 1.2977
0.7912 1.2977
0.7449 1.2977
0.7172 1.2977
0.6988 1.2977
PT
ED
M
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4
Fig. 1. Penrose tiling of 2D quasicrystal with 10-fold in the quasi-periodic (x1, x2)
AC
CE
plane: (a) 2D quasicrystal cell, (b) 2D quasicrystal solid
29
ACCEPTED MANUSCRIPT
Nxx z
x r
h
AN US
l
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θ
Nxx
Fig. 2. Schematic view of a 2D hexagonal QC cylindrical shell under axial
AC
CE
PT
ED
M
compression
30
ACCEPTED MANUSCRIPT
6
5
Pcr
3
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2
1
1.5
2.0
2.5
3.0
l/r
ED
1.0
M
0 0.5
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Case 1 Case 2: 1 Case 2: 2 Case 3: 1 Case 3: 2
4
PT
Fig. 3. Normalized Buckling loads Pcr of the QC cylindrical shell versus the
AC
CE
length-radius ratios l/r (l/h=10, γ1=γ2=1, m=n=1)
31
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0.200
r/h=5 r/h=10 r/h=20
0.175
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0.150
0.125 0.100
0.050 0.025
0.2
0.4
0.6
ED
M
0.000 0.0
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0.075
0.8
1.0
1.2
1.4
P
Fig. 4. Normalized vibration frequency Ω of the QC cylindrical shell for Case 1
AC
CE
m=n=1)
PT
versus the in-plane load P for different radius-thickness ratios r/h (l/r=1, γ1=1,
32
ACCEPTED MANUSCRIPT
0.12
r/h=5 r/h=10 r/h=20
0.10
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0.08
0.06
0.02
0.2
0.4
0.6
0.8
1.0
ED
M
0.00 0.0
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0.04
1.2
1.4
1.6
1.8
2.0
2.2
P
Fig. 5. Normalized vibration frequency Ω of the QC cylindrical shell for Case 1
AC
CE
m=n=1)
PT
versus the in-plane load P for different radius-thickness ratios r/h (l/r=2, γ1=1,
33
ACCEPTED MANUSCRIPT
0.06
r/h=5 r/h=10 r/h=20
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0.05
0.04
0.03
0.01
0.2
0.4
0.6
0.8
M
0.00 0.0
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0.02
1.0
1.2
1.4
1.6
1.8
P
ED
Fig. 6. Normalized vibration frequency Ω of the QC cylindrical shell for Case 1
AC
CE
m=n=1)
PT
versus the in-plane load P for different radius-thickness ratios r/h (l/r=3, γ1=1,
34
ACCEPTED MANUSCRIPT
0.6
P=0 P=1 P=2
0.5
CR IP T
0.4
0.3
0.1
0.0 2
3
4
5
ED
M
1
AN US
0.2
6
7
8
9
10
m
Fig. 7. Normalized vibration frequency Ω of the QC cylindrical shell for Case 1
AC
CE
γ1=1, n=1)
PT
versus the axial half-wave number m for different in-plane loads P (r/h=10, l/r=2,
35
ACCEPTED MANUSCRIPT
0.20
2/1=1
0.16
2/1=2
0.14
2/1=4
CR IP T
0.18
0.12 0.10
AN US
0.08 0.06 0.04
0.00 1
2
3
4
5
P
ED
0
M
0.02
PT
Fig. 8. Normalized vibration frequency Ω of the QC cylindrical shell for Case 2 versus the in-plane load P for different phason-phonon load ratios γ2/γ1 (r/h=10, γ1=1,
AC
CE
l/r=1, m=n=1)
36
ACCEPTED MANUSCRIPT
0.09
2/1=1
0.08
2/1=2
0.07
CR IP T
2/1=4
0.06
0.05
AN US
0.04 0.03 0.02 0.01
1
2
ED
0
M
0.00
3
4
5
P
Fig. 9. Normalized vibration frequency Ω of the QC cylindrical shell for Case 2
PT
versus the in-plane load P for different phason-phonon load ratios γ2/γ1 (r/h=10, γ1=1,
AC
CE
l/r=2, m=n=1)
37
ACCEPTED MANUSCRIPT
0.07
2/1=1
0.06
CR IP T
2/1=2 2/1=4
0.05
0.04
AN US
0.03
0.02
0.00 1
2
3
4
5
P
ED
0
M
0.01
PT
Fig. 10. Normalized vibration frequency Ω of the QC cylindrical shell for Case 2 versus the in-plane load P for different phason-phonon load ratios γ2/γ1 (r/h=10, γ1=1,
AC
CE
l/r=3, m=n=1)
38
ACCEPTED MANUSCRIPT
0.10
r/h=5 r/h=10 r/h=20
CR IP T
0.08
0.06
0.02
0.4
0.6
0.8
M
0.2
1.0
1.2
1.4
1.6
1.8
P
ED
0.00 0.0
AN US
0.04
PT
Fig. 11. Normalized vibration frequency Ω of the QC cylindrical shell for Case 2 versus the in-plane load P for different radius-thickness ratios r/h (l/r=2, γ1=1, γ2/γ1=4,
AC
CE
m=n=1)
39
ACCEPTED MANUSCRIPT
0.35
CR IP T
m=1 m=2 m=3
0.30
0.25
0.20
AN US
0.15
0.10
0.2
0.4
0.6
0.8
1.0
1.2
P
ED
0.00 0.0
M
0.05
PT
Fig. 12. Normalized vibration frequency Ω of the QC cylindrical shell for Case 2 versus the in-plane load P for different axial half-wave numbers m (l/r=1, γ1=1,
AC
CE
γ2/γ1=4, r/h=10, n=1)
40
ACCEPTED MANUSCRIPT
0.09
2/1=1
0.08
2/1=2
CR IP T
2/1=4
0.07
0.06 0.05
AN US
0.04 0.03 0.02
0.2
0.4
ED
0.0
M
0.01
0.6
0.8
1.0
P
Fig. 13. Normalized vibration frequency Ω of the QC cylindrical shell for Case 3
PT
versus the in-plane load P for different phason-phonon load ratios γ2/γ1 (r/h=10, γ1=1,
AC
CE
l/r=1, m=n=1)
41
ACCEPTED MANUSCRIPT
0.050
2/1=1
0.045
2/1=2
0.040
CR IP T
2/1=4
0.035
0.025
AN US
0.030
0.020 0.015
0.5
1.0
1.5
2.0
2.5
P
ED
0.005 0.0
M
0.010
PT
Fig. 14. Normalized vibration frequency Ω of the QC cylindrical shell for Case 3 versus the in-plane load P for different phason-phonon load ratios γ2/γ1 (r/h=10, γ1=1,
AC
CE
l/r=2, m=n=1)
42
ACCEPTED MANUSCRIPT
0.07
2/1=1
0.06
CR IP T
2/1=2 2/1=4
0.05
0.04
AN US
0.03
0.02
0.00 1
2
3
4
5
P
ED
0
M
0.01
PT
Fig. 15. Normalized vibration frequency Ω of the QC cylindrical shell for Case 3 versus the in-plane load P for different phason-phonon load ratios γ2/γ1 (r/h=10, γ1=1,
AC
CE
l/r=3, m=n=1)
43
ACCEPTED MANUSCRIPT
0.20 0.18
r/h=5 r/h=10 r/h=20
CR IP T
0.16 0.14
0.12 0.10
AN US
0.08 0.06 0.04
0.2
0.4
0.6
0.8
1.0
1.2
P
ED
0.00 0.0
M
0.02
PT
Fig. 16. Normalized vibration frequency Ω of the QC cylindrical shell for Case 3 versus the in-plane load P for different radius-thickness ratios r/h (l/r=1, γ1=1, γ2/γ1=2,
AC
CE
m=n=1)
44
ACCEPTED MANUSCRIPT
0.20
CR IP T
m=1 m=2 m=3
0.10
AN US
0.15
0.2
0.4
0.6
0.8
1.0
P
ED
0.00 0.0
M
0.05
PT
Fig. 17. Normalized vibration frequency Ω of the QC cylindrical shell for Case 3 versus the in-plane load P for different axial half-wave numbers m (l/r=1, γ1=1,
AC
CE
γ2/γ1=2, r/h=10, n=1)
45