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Thermo elasticity solution of sandwich circular plate with functionally graded core using generalized differential quadrature method
Composite Structures 136 (2016) 229–240
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Thermo elasticity solution of sandwich circular plate with functionally graded core using generalized differential quadrature method A. Alibeigloo ⇑ Mech. Eng. Dep., Faculty of Engineering, Tarbiat Modares University, Tehran 14115-143, Iran
a r t i c l e
i n f o
Article history: Available online 22 October 2015 Keywords: Sandwich Thermoelasticity GDQ FGM Circular plate
a b s t r a c t Bending analysis of sandwich circular plate with functionally grade core layer subjected to thermo-mechanical load is carried out using generalized differential quadrature (GDQ) method. The facing layers are made by metal and ceramic whereas the core layer is functionally graded materials composed of metal and ceramic material with distribution according to exponential function. Sandwich plate has various edges boundary conditions. Temperature distribution in three dimensions is obtained by solving heat conduction governing equation analytically. From combination of three dimensional governing equilibrium equations and constitutive relations, state space equations are derived. Applying GDQ method to the state space equations along the radial direction, semi-analytical solution can be obtained. After checking the convergence of the present approach, validation is carried out by comparing numerical results with the available results in open literature. Moreover, parametric study is presented to show the effects of the gradient direction, outer radius to thickness ratio, edges boundary conditions on the thermoelastic behavior of sandwich annular plates. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Sandwich plates are widely used in modern industry, especially in mechanical, nuclear reactors and aerospace industries due to its lightweight, high stiffness, high structural efficiency and strength. Due to the mismatch of thermal expansion coefficients as well as stiffness properties between the face sheets and the core, sandwich plates are susceptible to face sheet/core debonding, which is a major problem in sandwich construction. To overcome this problem, the concept of a functionally graded material (FGM) is being actively explored in sandwich panel design. Recently, many researchers have studied mechanical properties and behavior of FG sandwich structures. Based on higher order refined theory, Kant and Swaminathan [1] carried out static analysis of simply supported composite and sandwich plates using Navier’s technique. Anderson [2] presented an elasticity solution for static behavior of sandwich panel with orthotropic face sheets subjected to transverse loading. Based on higher-order refined theory, Swaminathan and Ragounadin [3] performed static analysis of simply supported anti-symmetric angle-ply composite and sandwich plates using Navier’s technique. Based on theory of elasticity, Li et al. [4] investigated free vibration of simply supported and clamped edges ⇑ Tel.: +98 21 82883991; fax: +98 21 82884909. E-mail address: [email protected] http://dx.doi.org/10.1016/j.compstruct.2015.10.012 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
sandwich plates with volume fraction distribution of constituents according to simple power law. Kant et al. [5] discussed semianalytically bending behavior of sandwich laminates using twopoint boundary value problem (BVP) governed by a set of linear first-order ordinary differential equations (ODEs) through the thickness. Pandit et al. [6] proposed an improved higher order zigzag theory to analysis of static behavior of laminated sandwich plate with soft compressible core. Brischetto [7] analyzed bending behavior of sandwich plates with material properties according to the Legendre polynomials using classical and mixed advanced models. Wang et al. [8] used direct displacement method to investigate free vibration of FGM circular plate with elastic simply supported and rigid slipping support edge. Based on higher order theory, Tu et al. [9] carried out bending and vibration analysis of laminated and sandwich composite plates using finite element method. Based on two-dimensional theory of elasticity, Pilipchuk et al. [10] presented an exact solution for bending of sandwich FG plate-like beams with simply supported boundary condition subjected to thermal load. Axisymmetric Bending of FGM circular plate was discussed by Wang et al. [11] using three-dimensional theory of elasticity as well as the direct displacement method. Based on higher-order shear deformation theory (HSDT), static analysis of FG sandwich plate was carried out by Abdelaziz et al. [12]. Based on higher order structural theory, Natarajan and Manickam [13] studied static and free vibration of sandwich
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
FGM plate using QUAD-8 shear flexible element developed. Neves et al. [14] used hyperbolic sine term for the in-plane displacements and a quadratic function of thickness coordinate for transverse displacement to present static analysis of FG sandwich plates. Based on Hamilton’s principle, Khalili et al. [15] investigated free vibration of sandwich plates with FG face sheets and assumption of material properties of FG face sheets to be temperaturedependent by a third-order function of temperature using. Alibeigloo [16] used three-dimensional theory of elasticity to analysis of a FG solid and an annular circular plate subjected to thermomechanical load with various boundary conditions. Wang and Shen [17] used a two-step perturbation technique to investigate large amplitude vibration and nonlinear bending of a sandwich plate with carbon nanotube-reinforced composite (CNTRC) face sheets resting on an elastic foundation in thermal environments. Based on two-dimensional shear deformation theory, Xiang et al. [18] employed meshless global collocation method to analysis of free vibration of sandwich plate with FG face sheets and homogeneous core. Loja et al. [19] used B-spline finite strip element models to study static and free vibration behavior of FG sandwich plate with assumption of material properties to be according to Mori–Tanaka formulation. Based on higher-order shear deformation theory, Neves et al. [20] employed a and meshless technique to analysis static, free vibration and buckling behaviors of simply supported isotropic and sandwich FG plates. Tounsi et al. [21] used a refined trigonometric shear deformation theory to study thermoelastic bending behavior of simply support FG sandwich plates. Shariyat and Alipour [22] used finite Taylor’s transform and the fourth order Runge–Kutta procedure to study dynamic behavior of circular sandwich plates with functionally graded face sheets/ cores. Alibeigloo [23] presented three dimensional thermoelastic solution of a simply supported sandwich panel with FG core using Fourier series expansions and state-space technique. Based on generalized displacement field of the Carrera Unified Formulation (CUF), including the Zig-Zag effect given by the Murakami’s function, Tornabene et al. [24] analyzed dynamic behavior of doubly curved shell structures using GDQ method. Based on twodimensional Unconstrained Third order Shear Deformation Theory (UTSDT), Viola et al. [25] performed static analysis of functionally graded conical shells and panels using GDQ method. Static analysis of FG circular plate made of magneto-electro-elastic (MEE) materials was carried out by Wang et al. [26] using three dimensional theory of elasticity. Based on Higher-order Shear Deformation Theories, Tornabene et al. [27] investigated recovery of through-thethickness transverse normal and shear strains and stresses due to static deformation of FG doubly-curved sandwich shell structures and shells of revolution using GDQ method. Mantari and Granados [28] used a new quasi-3D hybrid type HSDT with 6 unknowns which is based on a generalized formulation to investigate thermoelastic bending analysis of advanced composite sandwich plates. To the author’s knowledge, however, themoelastic analysis of sandwich circular plate with FGM core has not yet been reported. In this paper semi-analytical solution for sandwich circular plate with FGM core embedded with ceramic and metal face sheets subjected to thermos-mechanical load is investigated. 2. Analysis
Fig. 1. Geometry and coordinates of the laminated plate.
system r, h, z with the origin o, on the center of the bottom plane is employed to describe the plate behavior. Poisson’s ratio, m; is assumed to be constant and the material properties are assumed to vary according to exponential function of transverse coordinate as follow
E ¼ E0 eb1 ðzhm Þ
a ¼ a0 eb2 ðzhm Þ k ¼ k0 eb3 ðzhm Þ
ð1Þ
are material where b1 ¼ 1h ln EE0h , b2 ¼ 1h ln aa0h , b3 ¼ 1h ln kkh0 constants. Governing differential equation of the axisymmetric temperature field T(r, z) for the FGM and face sheets are respectively as
@2T f 1 @krf 1 @T f @ 2 T f þ þ þ 2 ¼0 krf @r r @r @r2 @z
ð2aÞ
kri @ @T i @2T i r þ kzi 2 ¼ 0 ði ¼ m; cÞ r @r @r @z
ð2bÞ
where m and c denotes metal and ceramic sheets, respectively Temperature boundary conditions are assumed to be
Tðr; 0Þ ¼ 0;
Tðr; hÞ ¼ T h ;
T i ðr i ; zÞ ¼ T i ðr o ; zÞ ¼ 0 ði ¼ m; f ; cÞ ð3aÞ
Relations for continuity of temperature and equilibrium of heat conduction at the interface of FGM and face sheets are
T c ðr; h hc Þ ¼ T f ðr; h hc Þ; kzc
T m ðr; hm Þ ¼ T f ðr; hm Þ
@T f @T c ¼ kzf ; @z z¼hhc @z z¼hhc
kzm
ð3bÞ
@T f @T m ¼ kzf @z z¼hm @z z¼hm
ð3cÞ
where subscripts f, c and m refers, respectively, to fields for the FGM, and ceramic and metal face sheets, T f , T c ; T m and kzm , kzc ; kzf are temperatures and thermal conductivity constants of the FGM core, ceramic and metal face sheets, and hm ¼ hc is the thickness of face sheets. For simplicity the following dimensionless quantities are introduced
i; s rz Þ ¼ ðr
ðri ; srz Þ Y 1 P1
i ¼ u
;k Þ ¼ ðk ; k Þ 1 ðk rj zj rj zj k1
ui P1 h
ði ¼ r; h; zÞ
E 0 Þ ¼ ðE; E0 Þ 1 ðE; Y1
i; a 0Þ ¼ ða
ðai ; a0 Þ
a1
2.1. Heat conduction problem Consider a sandwich circular plate made by FGM core and metal and ceramic face sheets which is subjected to uniform pressure at top surface as well as temperature difference between top and bottom surfaces. The outer and inner radius of the sandwich plate are ro and r i with uniform thickness h (Fig. 1). A cylindrical coordinate
a1 T j T j ¼ ðj ¼ c; f ; mÞ; P1
r ¼
r ro
z ¼
z h
k ¼ b h ðk ¼ 1; 2; 3Þ b k ð4Þ
where Y 1 ¼ 1 GPa, a1 ¼ 106 =K, k1 ¼ 2 factors.
W mK
T 1 ¼ 300 K are scale
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
Here P1 ¼ Yp01 is used for the case of purely mechanical loading, p0 ; and p1 ¼ a1 T h is used for the case of purely thermal loading, T h ; at the outer surface. The solution of Eq. (2a) is carried out by using separation of variables method, as the follow
T ¼ RðrÞ ZðzÞ
ð5Þ
where
C 0 ðknrÞ ¼ J 0 ðknr Þ
Enf ¼
0:5
e D
Substituting Eqs. (5), (1) and (4) into Eq. (2a) leads to the following heat conduction equation
! ! 2 b 1 @2Z @Z 1 @ 2 R 1 @R ¼ þ b þ 3 h Z @ z2 @z R @r2 r @r
ð6Þ
when the both sides are equal to a constant, k2 , which is called separation constant. It is noted that the sign of separation constant depends on the surface temperature boundary conditions. Considering this, Eq. (6) is converted to the following two ordinary differential equations in terms of only radial and or thickness variables for annular FGM core layer
@2 1 @ Rðr Þ þ k2 Rðr Þ ¼ 0 Rðr Þ þ r @r @r 2
ð7aÞ
2 @2 @ h zÞ þ b3 ZðzÞ k2 Zð ZðzÞ ¼ 0 @z ro @z2
ð7bÞ
Rðr Þ ¼ AJ 0 ðkr Þ þ BY 0 ðkrÞ ZðzÞ ¼ Ce
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b23 þ4ðkrho Þ z
0:5 b3
þ De
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b23 þ4ðkrho Þ z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h b23 þ4ðkn rho Þ J 0 ðkr i Þ m B¼ A D ¼ Ce Y 0 ðkr i Þ
ð9Þ
And the condition for non zero solution yields
J 0 ðkri ÞY 0 ðkÞ J 0 ðkÞY 0 ðkr i Þ ¼ 0
ð10Þ
Values of k are determined by solving Eq. (10). By using the inhomogeneous boundary condition and applying the orthogonality property, following temperature distribution for the plate 1 is obtained
0
h1f ¼
1 X 0:5 Enf @e n¼1
e
m h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b3 þ b23 þ4ðkn rho Þ z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b23 þ4ðkn rho Þ z
0:5 b3
e
h2f
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 b23 þ4ðkn rho Þ z A
C 20 ðkn Þ
2½C 1 ðkn Þ r i C 1 ðknr i Þ i h ii C 1 ðkn ÞC 1 ðkn Þ þ r 2i C 1 ðknri ÞC 1 ðknri Þ C 20 ðknr i Þ
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 X b23 þ4ðkn rho Þ z f @ 0:5 b3 þ ¼ Fn e n¼1
e
0:5 b3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 c Þ b23 þ4ðkn rho Þ zþð1h b23 þ4ðkn rho Þ A
C 0 ðknr Þ
ð13Þ
where e
0:5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 þ4ðkn h Þ2 h 3 þ b b ro 3
T fi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c Þ 2 þ4ðkn h Þ2 h 3 2 þ4ðkn h Þ2 0:5 b b ð1h b ro ro 3 3 e e ð14Þ
Applying superposition method to Eqs. (11) and (13) results in temperature distribution for the FGM layer as follow 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:5 b þ b2 þ4ðkn h Þ2 z X 3 ro f f 3 4 Ef þ F f e T f ¼ h1 þ h 2 ¼ n n n¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Þ 2 þ4ðkn h Þ2 z 3 2 þ4ðkn h Þ2 2 þ4ðkn h Þ2 b 0:5 b h ð1h b b c c ro ro ro f f 3 3 3 5 Ae þ Fn e @En e 0
ð8bÞ
here J0 ðkrÞ and Y 0 ðkrÞ are the Bessel functions of the first and second kind of zero order, respectively and A, B, C, D are unknown constants. This plate with two non-homogeneous boundary conditions is divided into two plates, each one having only one nonhomogeneous boundary condition which can be easily solved. Inhomogeneous boundary conditions for plate 1 and 2 are c Þ ¼ T m Þ ¼ T h1 ðr; 1 h and h2 ðr; h respectively. Relations fo fi between the constants of integration is determined by using the homogeneous boundary condition
T f 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ c b23 þ4ðkn rho Þ ð1h b23 þ4ðkn rho Þ 0:5 b3 h c e e
By using the same procedure, temperature distribution for the plate 2 is readily found to be
ð8aÞ
0:5 b3 þ
hh
D¼
F nf ¼
It is noted that separation constant, k is determined afterward from temperature boundary conditions. Eqs. (7a) and (7b) has analytical solution as the follow, respectively
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Þ b3 þ b23 þ4ðkn rho Þ ð1h c
ð12aÞ
ð12bÞ
kn
The left-hand side of Eq. (6) is a function of the variable r only and the right-hand side is a function of z only. Therefore, it is conclude that the only way that the above equation can hold is
J 0 ðknr i Þ Y 0 ðknr Þ Y 0 ðknr i Þ
C 0 ðknr Þ
ð15Þ
By substituting of Eqs. (1) and (4) into Eq. (2b) and applying separation of variables method, two ordinary differential equations in terms of only radial and thickness variables for annular face sheets can be derived
@2 1 @ Rðr Þ þ RðrÞ þ k2 Rðr Þ ¼ 0 r @r @r 2 ki
2 @2 h zÞ k2 Zð ZðzÞ ¼ 0 ði ¼ m; cÞ b @ z2
ð16aÞ
ð16bÞ
Temperature distribution for metal layer is computed by using m Þ ¼ T mo ; and one non-homogeneous boundary condition, h1 ðr; h applying the orthogonality property
Tm ¼
1 h kn X h kn r o km z Em ero km z C 0 ðknr Þ n e
ð17Þ
n¼1
where
Em n ¼
T m0 h kn r o k m hm
e
h kn
ero km hm
D
As before, temperature distribution for ceramic layer due to the one non-homogeneous boundary condition, h1 ðr; 1Þ ¼ T h can be found as
C 0 ðknrÞ
ð11Þ
hc1 ¼
1 h kn X h kn h kn Ecn ero kc z e2ro kc ð1hc Þ ero kc z C 0 ðknr Þ n¼1
ð18Þ
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
where
Ecn
¼
T h h kn
h kn
ero kc ero kc
D
1 h kn X h kn hkn F cn ero kc z e2ro kc hc ebkc z C 0 ðknr Þ
r z;z ¼ b1 r z z;z ¼ u
here
T ci h kn ro kc ð1hc Þ
e
h kn
h kn
e2ro kc ero km ð1hc Þ
D
ð20Þ
Temperature distribution for ceramic layer is determined by applying superposition method to Eqs. (18) and (19) 1 h kn X h kn h kn h kn Tc ¼ Ecn Ecn þ F cn ero kc z Ecn e2ro kc ð1hc Þ þ F cn e2ro kc ero kc z C 0 ðknr Þ
2 E0 h m h 1 1 r z;r þ u u u r r; r r; r r r ro 1 m ro 1 m2 r 2 0 b ðzhm Þ h E0 a 1 s rz þ he 2 T b ro 1 m
In this section thermo-elasticity of a sandwich circular plate with arbitrary edges boundary conditions is studied. Stress distributions in the FGM core layer can be written as
r r ¼
r h ¼
r s
¼ ðri ; srz Þe
b1 ðzhm Þ
ði ¼ r; h; zÞ
ð22Þ
From Eqs. (1), (4) and (22), non-dimensional thermo-elastic constitutive relations in term of displacements are introduced as
r r ¼
0 E h h r;r þ mu r þ mu z;z ð1 mÞ u ro ro ð1 þ mÞð1 2mÞ 0 b ðzhm Þ E0 a T e2 1 2m
srz ¼
0 E h r;z z;r þ u u 2ð1 þ mÞ ro
ð23Þ
Dimensionless governing equilibrium equation, in the absence of body force, for axisymmetric problems of FGM layer is
h h 1 r r;r þ srz;z þ ðr r r h Þ þ b1 srz ¼ 0 ro ro r
z ¼ 0 u
ð27bÞ
ð1 mÞ
r z þ
Since state Eq. (26) is not possible to solve analytically so, a semi-analytical procedure with the aids of DQ technique is use to solve it. In DQ method, the nth-order partial derivative of a continuous function f(x, z) with respect to x at a given point xi can be approximated to a linear sum of weighted function values at all of the discrete points as
ð28Þ
zi;z u
ð24Þ
ð25aÞ
!
N X ð1 þ mÞð1 2mÞ h m 1 ri þ rj ¼ r zi g ij u u ro 1 m ri E0 ð1 mÞ j¼1
þ
r;z ¼ u
And the pertinent relations for Simply (S) support, Clamped (C) support, Free (F) from supported at the r ¼ r i and r ¼ 1 edges are as;
r ¼ 0; S:r
0 m 1 0 b ðzhm Þ E0 a h E T þ m e2 u u r r; r r o 1 m2 r 1m
m
N X h 1 rzi þ rzj r zi;z ¼ b1 r zi s g ij s r o ri j¼1
h h 1 srz;r þ r z;z þ srz þ b1 r z ¼ 0 ro ro r
ð27aÞ
where N is the total number of discretization sampling points, and g nij are the xi -dependent weighted coefficients [29]. Applying Eq. (28) to Eq. (26) results in the following state equations at an arbitrary sampling point ri
0 E h h m ur;r þ m ur þ ð1 mÞuz;z ro ð1 þ mÞð1 2mÞ r o 0 b ðzhm Þ E0 a e2 T 1 2m
m E a E h r z þ z 0 2 h ur þ ur;r 0 0 eb2 ðzhm Þ T ro 1 m r ð1 mÞ 1m
m
n N X @f ðxi ; zÞ ¼ g nij f ðxj ; zÞ ði ¼ 1; . . . ; N n ¼ 1; . . . ; N 1Þ n @x x¼xi j¼1
0 h h E r h ¼ m ur;r þ ð1 mÞur þ muz;z ro ð1 þ mÞð1 2mÞ r o 0 b ðzhm Þ E0 a T e2 1 2m
r z ¼
ð26Þ
In-plane normal stresses in term of state variables are found from Eqs. (23) and (26) as the follow
2.2. Thermo-elastic problem for FGM layer
z 2ð1 þ mÞ h @u srz þ 0 r o @r E
srz;z ¼
ð21Þ
0 rz
h 1 rz;r rz þ s s r o r
ð1 þ mÞð1 2mÞ m h 1 @ z u r þ u r r r; 0 ð1 mÞ @r 1 m ro r E ð1 þ mÞa0 b2 ðzhm Þ þ e T 1m
r;z ¼ u
n¼1
0 i;
ð25cÞ
Using Eqs. (23) and (24) leads to the following state-space equations
ð19Þ
n¼1
F cn ¼
ð25bÞ
rz ¼ 0 r ¼ s F:r
By using the same procedure for the ceramic with the one nonc h Þ ¼ T ci ; temperhomogeneous boundary condition, h2 ðr ; 1 h f ature distribution will be
hc2 ¼
z @u ¼0 @r
r ¼ u z ¼ 0; C:u
!
0 b ðzhm Þ ð1 þ mÞa Ti e2 1m
N hX 2ð1 þ mÞ zj þ g u srzi 0 r o j¼1 ij E
2 N E0 h m X h zj þ g ij r r o 1 m j¼1 r o 1 m2 ! N N 0 a X 0 b ðzhm Þ ri 1 X h E u 2 1 s rj rj b rzi þ he 2 Ti g ij u g ij u r r 2i r 1 m o j¼1 j¼1
srzi;z ¼
ð29Þ
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
Similarly, from Eqs. (27a,b) and (28) it can be written
r ri ¼
m ð1 mÞ
r zi þ
N 0 h E m X rj g ij u uri þ 2 ro 1 m ri j¼1
!
Table 1 3 The deflection factor a ¼ wð0; 0:5Þ hb of simply supported FGM plate.
0 a 0 b ðzhm Þ E Ti e2 1m ð30aÞ
r hi ¼
m ð1 mÞ
r zi þ
N 0 m X h E 1 ri þ m g ij u rj u 2 ro 1 m ri j¼1
!
0 a 0 b ðzhm Þ E T i e2 1m ð30bÞ
where rki ¼ rk ðri ; zÞ ðk ¼ r; h; zÞ szri ¼ szr ðri ; zÞ uki ¼ uk ðri ; zÞ ðk ¼ r; zÞ. Assembly of Eq. (29) at all sampling points leads to the following global state equation in the matrix form
d df ¼ Gf df þ Pf T f dz
ð31Þ
where
z df ¼ f r
z u
r u
srz gT ; r z ¼ ½r z1 ; r z2 ; . . . ; r zN T
ð32Þ
Coefficient partitioned matrix Gf ; Pf and T f are described in Appendix A, and other sub-vectors in Eq. (31) are defined in the same manner as Eq. (32). Applying the boundary conditions at r ¼ ri ; 1 to the Eq. (31) results in the unique solution for the state variables, d. Taking into account the boundary conditions, the state-space Eq. (31) is converted to the following equation which has unique solution
d dfb ¼ Gfb dfb þ Pfb T f dz
ð33Þ
where the subscript, b, denotes that the state equation contains the boundary conditions and the matrix Gfb and Pfb according to each boundary condition type are defined in Appendix A.
k1
5 4 3 2 1 0 1 2 3 4 5
h b
¼ 0:1
h b
¼ 0:2
h b
¼ 0:3
Present
[11]
Present
[11]
Present
[11]
91.944 56.863 34.297 20.410 12.143 7.300 4.456 2.745 1.692 1.028 0.610
91.947 56.866 34.300 20.412 12.144 7.301 4.457 2.747 1.694 1.031 0.613
47.107 29.291 17.753 10.601 6.312 3.784 2.294 1.401 0.854 0.512 0.301
47.110 29.295 17.757 10.604 6.314 3.786 2.296 1.403 0.857 0.516 0.304
32.636 20.466 12.500 7.503 4.473 2.670 1.603 0.967 0.580 0.342 0.196
32.640 20.470 12.504 7.507 4.476 2.673 1.606 0.969 0.582 0.345 0.200
Analytical solution to Eq. (33) is
Z Þþ dfb ðzÞ ¼ eGfb z eGfb hm dfb ðh m
z
m h
m 6 z 6 1 h c h
eGfb g Pfb T f ðgÞdg
at ð34aÞ
2.3. Thermo-elastic problem of face sheets Governing state space differential equations containing edges boundary conditions for face sheets are derived from Eq. (33) by canceling b1 which their solution for metal and ceramic facing sheets are respectively as the follow
Z z at dmb ðzÞ ¼ eGmb z dmb ð0Þ þ eGmb g Pmb T m ðgÞdg
a. Radial normal stress
m 0 6 z 6 h
0
ð34bÞ
b. Circumferential normal stress
c. Transverse displacement Fig. 2. Convergence of the proposed method for sandwich annular plate with FGM core and S–S boundary condition.
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
" Gcb z
dcb ðzÞ ¼ e
e
m þh Þ Gcb ðh f
dcb h m þ hf þ
Z
z
þh h m f
e
Gcb g
# Pcb T c ðgÞdg
6 z 6 1 þh h m f
at
ð34cÞ
From Eqs. (28) and (30), in-plane stresses of face sheets in term of state variables are
E h m r ri ¼ r zi þ z j 2 h j uri þ r o 1 mj ð1 mj Þ r
mj
N X
!
rk g ik u
k¼1
j a j E T 1 mj ð35aÞ
r hi ¼
mj ð1 mj Þ
r zi þ
N j mj X h E 1 rj ri þ mj g ij u u 2 r o 1 mj r j¼1
!
j a j E T 1 mj
r z ¼ srz ¼ 0 at z ¼ 0
c. Transverse normal stress
e. Transverse displacement
p Þ ¼ M m ðdm ð0Þ þ Im Þ dm ðh
ð37aÞ
m þ h Þ ¼ M d ðh m Þ þ B I df ðh f f f f f
ð37bÞ
m þ h Þ þ Bc Ic dc ð1Þ ¼ Mc dc ðh f
ð37cÞ
Mm ¼ eGm hm ; Z
m h
Im ¼
M f ¼ eGf hf ;
Z Ic ¼
eGmb g :Pmb T m ðgÞdg;
1
m þh h f
Z If ¼
Bf ¼ eGf ðhm þhf Þ ;
M c ¼ e Gc hc ;
0
ð36aÞ ð36bÞ
a. Radial normal stress
Relation between state variable at the top and bottom surfaces of metal, FGM and ceramic layers are obtained from Eqs. (34a)–(34c) respectively as follow
where
ð35bÞ
where j ¼ c; m. Surface traction boundary condition at the top surface of the ceramic and bottom surface of the metal layers are, respectively
r z ¼ p0 ; srz ¼ 0 at z ¼ 1
2.4. Global transfer matrix and surface boundary conditions
m þh h f
m h
eGfb g :Pfb T f ðgÞdg
eGcb g Pcb T c ðgÞdg
b. Circumferential normal
Bc ¼ eGc
stress
d. Radial displacement
f. Temperature
Fig. 3. Effect of different lay up of sandwich plate on thermomechanical properties of sandwich circular plate with C–S boundary conditions.
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
Since the state variables at the interfaces of the metal, FGM and ceramic layers are continuous, so from Eqs. (37a)–(37c), following relation between the state variables at the top surface of ceramic and the bottom surface of metal is found
dc ð1Þ ¼ M dm ð0Þ þ B
ð38Þ
where N 1 ¼ Mc IF M f Mm and N2 ¼ Mc IF Mf M m Im þ Mc IF Bf If þ Bc Ic are the global transfer matrices. Imposing surface traction at the top and bottom surfaces of ceramic and metal layers respectively, Eqs. (36a) and (36b), to the Eq. (38) results in the following equations
8 9 P 0 > > > > > > > > < uz = > > > > :
2
M 11
6 6 M 21 ¼6 6 > ur > 4 M 31 > > ; 0 M 41 h
M 12
M 13
M 22
M 23
M 32
M 33
M 32
M 33
8 9 38 9 M14 > 0 > B1 > > > > > > > > > > > > > 7> < < = M24 7 uz B2 = 7 þ 7 > M34 5> B3 > > > ur > > > > > > > > > : > : > ; ; M34 0 0 B4
d. Transverse shear stress
uz ur
¼ 0
M 12 M 42
1
P0 0
B1 B4
ð40Þ
3. Numerical results and discussion In present work, sampling points with the following coordinates [29] are used;
ð39Þ
M 13 M 43
Then by using the obtained state variable at the bottom surface of the metal and using Eqs. (34a)–(34c), the state variables along the thickness of sandwich plate are determined. In-plane stresses are then computed by using Eqs. (35a) and (35b).
xj ¼
where Mij and Bi are square and column sub-matrixes with dimension depending on the case of edge boundary conditions. By solving Eq. (39), displacements at the bottom surface of metal layer are computed as follow
a. Radial normal stress
1 cos
ðj 1Þp r o r i N1 2
j ¼ 1; 2; . . . ; N
The material properties of FGM core layer is made by Metal/ Ceramic with the following material properties
Em ¼ 227:24 GPa; am ¼ 15 106 =K; km ¼ 25 W=mK for Metal Ec ¼ 125:83 GPa; ac ¼ 10 106 =K; kc ¼ 2:09 W=mK for Ceramic
b. Circumferential
normal stress
e. Radial displacement
f. Transverse displacement
Fig. 4. Effect of edges boundary conditions on the through-thickness distribution of displacements and stresses at the mid radius, r ¼ ro 2þri for hybrid annular FGM plate.
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
a. Radial normal stress
b. Circumferential normal stress
c. Transverse normal stress
d. Transverse shear stress
e. Radial displacement
f. Transverse displacement
g. Temperature Fig. 5. Influence of thermal load on the stresses, displacements distributions across the thickness of FGM annular plate with c–c condition.
m ¼ 0:3; T h ¼ 350 K;
hFG ¼5 hm
To show convergence of the present formulation, numerical results of dimensionless deflection as well as radial and circumferential normal stresses are computed and plotted in Fig. 2. As the figures show non-dimensional radial stresses is independent of increasing sampling points. Also it is observed that distribution of dimensionless deflection and circumferential stress along the thickness direction at mid radius of sandwich plate approaches
to a specific constant value at about 14 discrete points which confirms good convergence of the method. Since thermoelastic response of circular sandwich plate has not yet been reported in the literature, so in order to validate the present approach, a comparison study for the thermoelastic behavior of circular plate without face sheets is carried out. Numerical result of deflection factor (a) for circular FGM plate subjected to mechanical load is computed and along with the reported results by Yun et al. [11] are tabulated in Table 1. Regardless of the amount of thickness-toradius ratio and gradient index, deflection factor of present
A. Alibeigloo / Composite Structures 136 (2016) 229–240
formulation is smaller than that of Ref. [11]. According to the table in isotropic material, k1 ¼ 0, the results of the present approach is exactly the same as the results obtained by Yun et al. [11]. Furthermore it can be observed that by increasing the gradient index as well as the thickness to radius ratio, difference of two results increases. It is noted that the discrepancy of the two results is due to the division of FGM layer to the some fictitious isotropic layers with different thermoelastic constants in Ref. [11]. Effects of lay-up on thermo-elastic behavior of the sandwich annular plate subjected to thermal load and with S–C boundary conditions are considered in Fig. 3a–f. According to the figures in the sequence of metal, FGM and ceramic layup, M/FGM/C, through the thickness distribution of stresses, displacement and temperature at a given point are always smaller in magnitude than those at the corresponding points in the sequence of ceramic, FGM and metal, C/FGM/M, layup. In addition, From Fig. 3a and b it is concluded that the interface in-plane stresses are continuous. Through thickness distribution of stresses and displacements of sandwich annular plate with various edges condition are presented in Fig. 4a–e. From Fig. 4a–c it is seen that all of stresses distribution in S–S boundary condition, compared with the other edges conditions, has minimum value and in C–C boundary conditions has maximum value. Moreover, it can be concluded that stresses distribution in FGM layer is nonlinear function of transverse coordinate whiles it is linear in facing sheets and this is due to the variation of material properties in FGM layer according to the exponential function of transverse coordinate. Fig. 4d depicts the effect of edges boundary conditions on through the thickness distribution of radial
a. Radial normal stress
c.Transverse normal stresst
e. Radial displacement
237
displacements. According to the figure this distribution for the plate with C–C boundary conditions is minimum whereas it is maximum for the plate with S–S boundary condition. As the Fig. 4e shows, distribution of transverse displacement along the thickness direction is maximum for C–F boundary condition whereas it is minimum value for C–C boundary condition. Fig. 5a–g depicts through-thickness distributions of the dimensionless stresses, displacement and temperature with various surface temperature difference conditions. As the Fig. 5a and b show dimensionless radial and circumferential stresses increases by increasing surface temperature difference whiles transverse normal stress (Fig. 5c) decrease. Besides, it is concluded that distribution of radial and circumferential normal stresses at top surface is more affected by temperature difference than that at the bottom surface whereas this is converse for the transverse normal stress rz ; (Fig. 5d) increase by distribution. Transverse shear stress, s increasing the temperature difference in the region nearly bellow the mid surface and then decreases above the mid surface of the sandwich plate. From Fig. 5e and f it can be observed that radial displacement increases and transverse displacement decreases when surface temperature difference increases. Through the thickness temperature distribution is represented in Fig. 5g. According to the figure and as expected, increase surface temperature difference causes to increase temperature gradient along the thickness direction. Influence of surface tractions on through the thickness distribution of displacements and stresses for sandwich circular plate with FGM core is depicted in Fig. 6a–f. As the Fig. 6a shows, the neutral surface does not coincide with the mid
b. Circumferential normal stress
d. Transverse shear stress
f. Transverse displacement
Fig. 6. Influence of mechanical load on the stresses, displacements distributions across the thickness of FGM annular plate with c–c condition.
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Table 2 Non dimensional thermal stresses sandwich circular plate with S–S, P = 0, T 1 ¼ 300 K T 2 ¼ 350 K, r ¼ 0:5ðr o þ r i Þ, hhFG ¼ 5. m h
h
h
h
ro ri
ro h
r r
2
10 20 30 40 50 60 70
50.7345 61.3998 64.6837 66.3456 67.3640 68.0269 68.4583
0.4575 0.3792 0.1783 0.0638 0.0215 0.0064 0.0013
92.0854 13.2075 50.9592 70.9935 83.5109 91.9009 97.6917
11.5524 6.9198 6.0386 4.9821 4.0491 3.3167 2.7615
3
10 20 30 40 50 60 70
49.3851 64.7846 70.3959 73.3464 75.0967 76.1712 76.8388
0.3397 0.3017 0.0738 0.0114 0.0038 0.0064 0.0058
27.7063 48.2287 75.7091 90.1713 98.7314 103.9813 107.2398
14.6671 9.6147 7.1723 5.4440 4.2709 3.4691 2.9060
10 20 30 40 50 60 70
47.3694 64.7281 70.7991 73.8992 75.6593 76.6809 77.2741
0.3934 0.2253 0.0390 0.0019 0.0090 0.0085 0.0067
3.5841 60.2734 83.3312 95.2430 102.0131 105.9569 108.2723
17.3542 10.9166 7.7273 5.7359 4.4685 3.6302 3.0499
4
2
r z
2
r h
2
srz
2
thickness of the FGM core and it is shifted toward the stiffer surface. Moreover it is seen that stresses distribution satisfies surfaces boundary condition. According to the figures and as expected increase mechanical load causes to increase stresses as well as displacements. Effect of outer radius to thickness ratio, rho , on thermal stresses at mid radius and mid surface of plate with S-S boundary conditions for various outer radius to inner radius ratio, rro ; are i
computed numerically and tabulated in Table 2. It is shown in Table 2 that increase rho causes to increase radial and circumferential stresses and to decrease the transverse normal and shear stresses. In addition, it is observed that when rroi increases, all of stresses except to transverse shear stress decreases. 4. Conclusion In this investigation axisymmetric thermoelastic analysis of sandwich circular plate with FGM core imbedded in metal and ceramic facing sheets with various boundary conditions is carried out using semi-analytical state space differential quadrature method. Analytical solution along the thickness direction is performed by using state space technique and approximate solution in the radial direction is obtained by applying one-dimensional differential quadrature method. This method overcomes the difficulty conformed by the traditional state space method when applied to analyses the plate with non-simply supported boundary conditions. From parametric study it was concluded that in the sequence of metal, FGM and ceramic layup, M/FGM/C, through the thickness distribution of stresses, displacement and temperature at a given point are always smaller in magnitude than those at the corresponding points in the sequence of ceramic, FGM and metal, C/FGM/M, layup. Also numerical results predicted the continuity of interface in-plane stresses. The neutral surface of the FGM is not at the mid-surface but depends on the through-thickness variation of Young’s modulus. Stress field in S–S boundary condition has minimum value and in C–C boundary conditions it is maximum value. Increasing the surface temperature difference causes to increase in-plane normal stresses and to decrease transverse normal stress. It was concluded that radial displacement increases and transverse displacement decreases when surface temperature difference increases.
Appendix A 3 1 IN 0N 0N rho g ij þ 1r ij b 7 6 7 6 1 7 6 ð1þmÞð12mÞ h m 0N 0N ro 1m g ij þ r ij 7 6 E0 ð1mÞ IN 7 6 Gf ¼ 6 7 2ð1þmÞ 7 6 h 0N ro g ij 0N 7 6 E 0 7 6 5 4 2 2 E0 h 2 1 1 b rho 1m m g ij 0N g g þ I 1 N ij ro r ij ij r ij 1m2 2
ði; j ¼ 1; . . . ; NÞ
1
where
r ij
¼
1 ri
0
i ¼ j i–j
i; j ¼ 1; . . . ; N
2 3 T1 6 7 6 T2 7 6 7 6 7 6 : 7 7 T f ¼ 6 6 : 7 6 7 6 7 6 : 7 4 5 T N
3
2
0N 7 6 6 ð1 þ mÞIN 7 a 0 b2 ðzhm Þ 7 6 Pf ¼ 6 e 7 7 1m 6 0 N 5 4 0 h g T E r o ij S–S –FGM layer
2
1 IðN2Þ b 6 6 ð1þmÞð12mÞ 6 E0 ð1mÞ IðN2Þ 6 6 0NðN2Þ Gfb ¼ 6 6 6 4 rho 1m m g Tss
0ðN2Þ 0ðN2Þ
0ðN2ÞN rho
rho g Tss
m
1m
2 h ro
0NðN2Þ
rho ðg ss þ ass Þ
ðg ss þ ass Þ
0ðN2ÞN
0N
2ð1þmÞ IN E
E 0 1m2
0
ðkss Þ
1 IN b
ði; j ¼ 1; . . . ; NÞ 2
3
0ðN2Þ
7 6 6 ð1 þ mÞIðN2Þ 7 a 0 b2 ðzhm Þ 7 6 Pfb ¼ 6 e ; 7 6 0NðN2Þ 7 1 m 5 4 0 h g T E r o ss
g ssij ¼ g ij i ¼ 2; . . . ;
N 1 J ¼ 1; . . . ; N where
1 1 ass ¼ 0ðN2Þ1 0ðN2Þ1 r ss r ssij (1 i¼j ri i; j ¼ 2; . . . ; N 1 ¼ 0 i–j kss ¼ g 2ss þ f ss þ f ssn þ ¼ 1; . . . ; N;
2 1 þ g ss2 þ bss ; r ss2
f ssij ¼ g i1 g 1j i; j ¼ 1; . . . ; N
f ssnij ¼ g iN g Nj i; j ¼ 1; . . . ; N;
¼
8 2 < 1 :
ri
mg i1 r1
i¼j
2 1 r ss2ij
i; j ¼ 1; . . . ; N
i–j
0
bssij ¼
g 2ssij ¼ g 2ij i; j
0NðN2Þ N1
C–C –FGM layer
mg iN rN
; N1
g ss2ij ¼
g ij i; j ¼ 1; . . . ; N ri
3 7 7 7 7 7 7 7 7 5
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
2
1 IðN2Þ b 6 6 ð1þmÞð12mÞ 6 6 E0 ð1mÞ IðN2Þ 6 6 0ðN2Þ Gfb ¼ 6 6 6 4 rh0 1m m g cc
rh0 ðg cc þ ð1r Þcc Þ
0ðN2Þ
0ðN2Þ h m r0 1m g cc þ 1r cc
rh0 g cc
0ðN2Þ
2ð1þmÞ IðN2Þ E
0ðN2Þ
2 h r0
0ðN2Þ
E 0 1m2
g cc2 g 2cc þ
0ðN2Þ 0
1 2
1 IðN2Þ b
r cc
3
2
7 7 7 7 7 7 7 7 7 5
6 6 6 6 6 6 Gfb ¼ 6 6 6 4
ði;j ¼ 1;::; NÞ
g 2ccij ¼ g 2ij i; j ¼ 2; . . . ; N 1
m
ðg g þ g iN g Nj Þ i; j ¼ 2; . . . ; N 1; 1 2m i1 1j 8 2 < 1 i¼j ri i; j ¼ 2; . . . ; N 1 ¼ : 0 i–j
g 0cfij
2 1 r ccij
¼ g ij i; j ¼ 2; . . . ; N 1;
acf ¼
kcf ¼ g cf 2 g 2cf þ f cf þ ccf þ dcf ;
0
1 IðN1Þ b
#
0ðn2Þðn1Þ 01ðn1Þ
g cf 2ij ¼
g 2cfij ¼ g 2ij i ¼ 1; . . . ; N 1 J ¼ 2; . . . ; N; "
0ðN2Þ
0ðN2ÞðN1Þ
rho ðg cs þ acs Þ 0ðN2ÞðN1Þ 2ð1þmÞ IðN1Þ E 0
1 IðN1Þ b
3 7 7 7 7 7 7 7 7 5
ði; j ¼ 1; . . . ; NÞ 3 0ðN2Þ 7 6 6 ð1 þ mÞIðN2Þ 7 a 0 b2 ðzhm Þ 7 6 Pfb ¼ 6 e ; 7 6 0ðN1ÞðN2Þ 7 1 m 5 4 0 h g T E r o cs 2
g csij ¼ g ij i ¼ 2; . . . ; N 1 J ¼ 2; . . . ; N g 2csij ¼ g 2ij i; j ¼ 2; . . . ; N; (1
i¼j 1 1 r i; j ¼ 2; ::; N 1 0ðN2Þ1 ; ¼ i acs ¼ r cs r csij 0 i–j i¼j i–j
g ij i; j ¼ 1; . . . ; N; ri
i; j ¼ 2; . . . ; N;
f csij ¼
m2 1 2m
1 g cs2ij f csij r cs2ij
C–F –FGM layer
0ðN1Þ
0ðN1Þ 2ð1þmÞ IðN1Þ E
7 7 7 7 7 7 7 7 7 5
1 rN
g ij i ¼ 1; . . . ; N 1 J ri
f cfij ¼ g iN g Nj i
¼ 1; . . . ; N 1 J ¼ 2; . . . ; N
1 IðN2Þ b
kcsij ¼ g 2csij þ
g cf
0ðN1Þ 2 E0 h ðkcf Þ ro 1m2
r cf
¼ 2; . . . ; N
6 6 ð1þmÞð12mÞ I 0ðN2Þ rho 1m m ðg cs þ acs Þ 6 E0 ð1mÞ ðN2Þ 6 6 6 rho g Tcs 0ðN1Þ Gfb ¼ 6 0ðN1ÞðN2Þ 6 4 2 E0 h rho 1m m g Tcs 0ðN1ÞðN2Þ ðkcs Þ ro 1m2
g ss2ij ¼
m
1m
þ hcf
rho g cf þ
( 1 i¼j 1 ri i; j ¼ 1; . . . ; N 1 g cfij ¼ g ij i; j ¼ 1; . . . ; N 1; ¼ r cf 0 i–j
C–S –FGM layer
8 < 1 2 1 ri ¼ r cs2 : 0
rho
rho g 0cf
g0 1m cf m
"
2
2
0ðN1Þ
rho
3 0ðN1ÞðN2Þ 7 6 ð1 þ mÞI ðN2Þ 7 6 7 6 7 a0 eb2 ðzhm Þ 0 Pfb ¼ 6 1ðN2Þ 7 1m 6 7 6 0 ðN1ÞðN2Þ 5 4 0 h g E r o cf 1
g ij g cc2ij ¼ i; j ¼ 2; . . . ; N 1; ri ( 1 i¼j 1 ri i; j ¼ 2; . . . ; N 1 ¼ r ccij 0 i–j f ccij ¼
0ðN1Þ
0ðN1Þ
2
0ðN2Þ 7 6 6 ð1 þ mÞIðN2Þ 7 a 0 b2 ðzhm Þ 7 6 Pfb ¼ 6 e ; 7 7 6 1 m 0 ðN2Þ 5 4 0 h g E ro cc g ccij ¼ g ij i; j ¼ 2; . . . ; N 1;
acf
0ðN1Þ
1 3
ði; j ¼ 1; . . . ; NÞ
3
2
1 IðN1Þ b
g cs2ij ¼
g ij i; j ¼ 2; . . . ; N ri
g i1 g 1j g i1 g 1j i; j ¼ 2; . . . ; N
ccf ¼
01ðN2Þ 1 2 r cf
g cf 1ij ¼ g ij
# 0ðN2Þ1
mg jN ; dcf ¼ 0ðn1ÞðN2Þ ; j ¼ 1; . . . ; N 1 rN
i ¼ 1; . . . ; N 1; j ¼ 2; . . . ; N 1:
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Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Thermo elasticity solution of sandwich circular plate with functionally graded core using generalized differential quadrature method A. Alibeigloo ⇑ Mech. Eng. Dep., Faculty of Engineering, Tarbiat Modares University, Tehran 14115-143, Iran
a r t i c l e
i n f o
Article history: Available online 22 October 2015 Keywords: Sandwich Thermoelasticity GDQ FGM Circular plate
a b s t r a c t Bending analysis of sandwich circular plate with functionally grade core layer subjected to thermo-mechanical load is carried out using generalized differential quadrature (GDQ) method. The facing layers are made by metal and ceramic whereas the core layer is functionally graded materials composed of metal and ceramic material with distribution according to exponential function. Sandwich plate has various edges boundary conditions. Temperature distribution in three dimensions is obtained by solving heat conduction governing equation analytically. From combination of three dimensional governing equilibrium equations and constitutive relations, state space equations are derived. Applying GDQ method to the state space equations along the radial direction, semi-analytical solution can be obtained. After checking the convergence of the present approach, validation is carried out by comparing numerical results with the available results in open literature. Moreover, parametric study is presented to show the effects of the gradient direction, outer radius to thickness ratio, edges boundary conditions on the thermoelastic behavior of sandwich annular plates. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Sandwich plates are widely used in modern industry, especially in mechanical, nuclear reactors and aerospace industries due to its lightweight, high stiffness, high structural efficiency and strength. Due to the mismatch of thermal expansion coefficients as well as stiffness properties between the face sheets and the core, sandwich plates are susceptible to face sheet/core debonding, which is a major problem in sandwich construction. To overcome this problem, the concept of a functionally graded material (FGM) is being actively explored in sandwich panel design. Recently, many researchers have studied mechanical properties and behavior of FG sandwich structures. Based on higher order refined theory, Kant and Swaminathan [1] carried out static analysis of simply supported composite and sandwich plates using Navier’s technique. Anderson [2] presented an elasticity solution for static behavior of sandwich panel with orthotropic face sheets subjected to transverse loading. Based on higher-order refined theory, Swaminathan and Ragounadin [3] performed static analysis of simply supported anti-symmetric angle-ply composite and sandwich plates using Navier’s technique. Based on theory of elasticity, Li et al. [4] investigated free vibration of simply supported and clamped edges ⇑ Tel.: +98 21 82883991; fax: +98 21 82884909. E-mail address: [email protected] http://dx.doi.org/10.1016/j.compstruct.2015.10.012 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
sandwich plates with volume fraction distribution of constituents according to simple power law. Kant et al. [5] discussed semianalytically bending behavior of sandwich laminates using twopoint boundary value problem (BVP) governed by a set of linear first-order ordinary differential equations (ODEs) through the thickness. Pandit et al. [6] proposed an improved higher order zigzag theory to analysis of static behavior of laminated sandwich plate with soft compressible core. Brischetto [7] analyzed bending behavior of sandwich plates with material properties according to the Legendre polynomials using classical and mixed advanced models. Wang et al. [8] used direct displacement method to investigate free vibration of FGM circular plate with elastic simply supported and rigid slipping support edge. Based on higher order theory, Tu et al. [9] carried out bending and vibration analysis of laminated and sandwich composite plates using finite element method. Based on two-dimensional theory of elasticity, Pilipchuk et al. [10] presented an exact solution for bending of sandwich FG plate-like beams with simply supported boundary condition subjected to thermal load. Axisymmetric Bending of FGM circular plate was discussed by Wang et al. [11] using three-dimensional theory of elasticity as well as the direct displacement method. Based on higher-order shear deformation theory (HSDT), static analysis of FG sandwich plate was carried out by Abdelaziz et al. [12]. Based on higher order structural theory, Natarajan and Manickam [13] studied static and free vibration of sandwich
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FGM plate using QUAD-8 shear flexible element developed. Neves et al. [14] used hyperbolic sine term for the in-plane displacements and a quadratic function of thickness coordinate for transverse displacement to present static analysis of FG sandwich plates. Based on Hamilton’s principle, Khalili et al. [15] investigated free vibration of sandwich plates with FG face sheets and assumption of material properties of FG face sheets to be temperaturedependent by a third-order function of temperature using. Alibeigloo [16] used three-dimensional theory of elasticity to analysis of a FG solid and an annular circular plate subjected to thermomechanical load with various boundary conditions. Wang and Shen [17] used a two-step perturbation technique to investigate large amplitude vibration and nonlinear bending of a sandwich plate with carbon nanotube-reinforced composite (CNTRC) face sheets resting on an elastic foundation in thermal environments. Based on two-dimensional shear deformation theory, Xiang et al. [18] employed meshless global collocation method to analysis of free vibration of sandwich plate with FG face sheets and homogeneous core. Loja et al. [19] used B-spline finite strip element models to study static and free vibration behavior of FG sandwich plate with assumption of material properties to be according to Mori–Tanaka formulation. Based on higher-order shear deformation theory, Neves et al. [20] employed a and meshless technique to analysis static, free vibration and buckling behaviors of simply supported isotropic and sandwich FG plates. Tounsi et al. [21] used a refined trigonometric shear deformation theory to study thermoelastic bending behavior of simply support FG sandwich plates. Shariyat and Alipour [22] used finite Taylor’s transform and the fourth order Runge–Kutta procedure to study dynamic behavior of circular sandwich plates with functionally graded face sheets/ cores. Alibeigloo [23] presented three dimensional thermoelastic solution of a simply supported sandwich panel with FG core using Fourier series expansions and state-space technique. Based on generalized displacement field of the Carrera Unified Formulation (CUF), including the Zig-Zag effect given by the Murakami’s function, Tornabene et al. [24] analyzed dynamic behavior of doubly curved shell structures using GDQ method. Based on twodimensional Unconstrained Third order Shear Deformation Theory (UTSDT), Viola et al. [25] performed static analysis of functionally graded conical shells and panels using GDQ method. Static analysis of FG circular plate made of magneto-electro-elastic (MEE) materials was carried out by Wang et al. [26] using three dimensional theory of elasticity. Based on Higher-order Shear Deformation Theories, Tornabene et al. [27] investigated recovery of through-thethickness transverse normal and shear strains and stresses due to static deformation of FG doubly-curved sandwich shell structures and shells of revolution using GDQ method. Mantari and Granados [28] used a new quasi-3D hybrid type HSDT with 6 unknowns which is based on a generalized formulation to investigate thermoelastic bending analysis of advanced composite sandwich plates. To the author’s knowledge, however, themoelastic analysis of sandwich circular plate with FGM core has not yet been reported. In this paper semi-analytical solution for sandwich circular plate with FGM core embedded with ceramic and metal face sheets subjected to thermos-mechanical load is investigated. 2. Analysis
Fig. 1. Geometry and coordinates of the laminated plate.
system r, h, z with the origin o, on the center of the bottom plane is employed to describe the plate behavior. Poisson’s ratio, m; is assumed to be constant and the material properties are assumed to vary according to exponential function of transverse coordinate as follow
E ¼ E0 eb1 ðzhm Þ
a ¼ a0 eb2 ðzhm Þ k ¼ k0 eb3 ðzhm Þ
ð1Þ
are material where b1 ¼ 1h ln EE0h , b2 ¼ 1h ln aa0h , b3 ¼ 1h ln kkh0 constants. Governing differential equation of the axisymmetric temperature field T(r, z) for the FGM and face sheets are respectively as
@2T f 1 @krf 1 @T f @ 2 T f þ þ þ 2 ¼0 krf @r r @r @r2 @z
ð2aÞ
kri @ @T i @2T i r þ kzi 2 ¼ 0 ði ¼ m; cÞ r @r @r @z
ð2bÞ
where m and c denotes metal and ceramic sheets, respectively Temperature boundary conditions are assumed to be
Tðr; 0Þ ¼ 0;
Tðr; hÞ ¼ T h ;
T i ðr i ; zÞ ¼ T i ðr o ; zÞ ¼ 0 ði ¼ m; f ; cÞ ð3aÞ
Relations for continuity of temperature and equilibrium of heat conduction at the interface of FGM and face sheets are
T c ðr; h hc Þ ¼ T f ðr; h hc Þ; kzc
T m ðr; hm Þ ¼ T f ðr; hm Þ
@T f @T c ¼ kzf ; @z z¼hhc @z z¼hhc
kzm
ð3bÞ
@T f @T m ¼ kzf @z z¼hm @z z¼hm
ð3cÞ
where subscripts f, c and m refers, respectively, to fields for the FGM, and ceramic and metal face sheets, T f , T c ; T m and kzm , kzc ; kzf are temperatures and thermal conductivity constants of the FGM core, ceramic and metal face sheets, and hm ¼ hc is the thickness of face sheets. For simplicity the following dimensionless quantities are introduced
i; s rz Þ ¼ ðr
ðri ; srz Þ Y 1 P1
i ¼ u
;k Þ ¼ ðk ; k Þ 1 ðk rj zj rj zj k1
ui P1 h
ði ¼ r; h; zÞ
E 0 Þ ¼ ðE; E0 Þ 1 ðE; Y1
i; a 0Þ ¼ ða
ðai ; a0 Þ
a1
2.1. Heat conduction problem Consider a sandwich circular plate made by FGM core and metal and ceramic face sheets which is subjected to uniform pressure at top surface as well as temperature difference between top and bottom surfaces. The outer and inner radius of the sandwich plate are ro and r i with uniform thickness h (Fig. 1). A cylindrical coordinate
a1 T j T j ¼ ðj ¼ c; f ; mÞ; P1
r ¼
r ro
z ¼
z h
k ¼ b h ðk ¼ 1; 2; 3Þ b k ð4Þ
where Y 1 ¼ 1 GPa, a1 ¼ 106 =K, k1 ¼ 2 factors.
W mK
T 1 ¼ 300 K are scale
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Here P1 ¼ Yp01 is used for the case of purely mechanical loading, p0 ; and p1 ¼ a1 T h is used for the case of purely thermal loading, T h ; at the outer surface. The solution of Eq. (2a) is carried out by using separation of variables method, as the follow
T ¼ RðrÞ ZðzÞ
ð5Þ
where
C 0 ðknrÞ ¼ J 0 ðknr Þ
Enf ¼
0:5
e D
Substituting Eqs. (5), (1) and (4) into Eq. (2a) leads to the following heat conduction equation
! ! 2 b 1 @2Z @Z 1 @ 2 R 1 @R ¼ þ b þ 3 h Z @ z2 @z R @r2 r @r
ð6Þ
when the both sides are equal to a constant, k2 , which is called separation constant. It is noted that the sign of separation constant depends on the surface temperature boundary conditions. Considering this, Eq. (6) is converted to the following two ordinary differential equations in terms of only radial and or thickness variables for annular FGM core layer
@2 1 @ Rðr Þ þ k2 Rðr Þ ¼ 0 Rðr Þ þ r @r @r 2
ð7aÞ
2 @2 @ h zÞ þ b3 ZðzÞ k2 Zð ZðzÞ ¼ 0 @z ro @z2
ð7bÞ
Rðr Þ ¼ AJ 0 ðkr Þ þ BY 0 ðkrÞ ZðzÞ ¼ Ce
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b23 þ4ðkrho Þ z
0:5 b3
þ De
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b23 þ4ðkrho Þ z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h b23 þ4ðkn rho Þ J 0 ðkr i Þ m B¼ A D ¼ Ce Y 0 ðkr i Þ
ð9Þ
And the condition for non zero solution yields
J 0 ðkri ÞY 0 ðkÞ J 0 ðkÞY 0 ðkr i Þ ¼ 0
ð10Þ
Values of k are determined by solving Eq. (10). By using the inhomogeneous boundary condition and applying the orthogonality property, following temperature distribution for the plate 1 is obtained
0
h1f ¼
1 X 0:5 Enf @e n¼1
e
m h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b3 þ b23 þ4ðkn rho Þ z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b23 þ4ðkn rho Þ z
0:5 b3
e
h2f
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 b23 þ4ðkn rho Þ z A
C 20 ðkn Þ
2½C 1 ðkn Þ r i C 1 ðknr i Þ i h ii C 1 ðkn ÞC 1 ðkn Þ þ r 2i C 1 ðknri ÞC 1 ðknri Þ C 20 ðknr i Þ
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 X b23 þ4ðkn rho Þ z f @ 0:5 b3 þ ¼ Fn e n¼1
e
0:5 b3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 c Þ b23 þ4ðkn rho Þ zþð1h b23 þ4ðkn rho Þ A
C 0 ðknr Þ
ð13Þ
where e
0:5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 þ4ðkn h Þ2 h 3 þ b b ro 3
T fi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c Þ 2 þ4ðkn h Þ2 h 3 2 þ4ðkn h Þ2 0:5 b b ð1h b ro ro 3 3 e e ð14Þ
Applying superposition method to Eqs. (11) and (13) results in temperature distribution for the FGM layer as follow 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0:5 b þ b2 þ4ðkn h Þ2 z X 3 ro f f 3 4 Ef þ F f e T f ¼ h1 þ h 2 ¼ n n n¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Þ 2 þ4ðkn h Þ2 z 3 2 þ4ðkn h Þ2 2 þ4ðkn h Þ2 b 0:5 b h ð1h b b c c ro ro ro f f 3 3 3 5 Ae þ Fn e @En e 0
ð8bÞ
here J0 ðkrÞ and Y 0 ðkrÞ are the Bessel functions of the first and second kind of zero order, respectively and A, B, C, D are unknown constants. This plate with two non-homogeneous boundary conditions is divided into two plates, each one having only one nonhomogeneous boundary condition which can be easily solved. Inhomogeneous boundary conditions for plate 1 and 2 are c Þ ¼ T m Þ ¼ T h1 ðr; 1 h and h2 ðr; h respectively. Relations fo fi between the constants of integration is determined by using the homogeneous boundary condition
T f 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ c b23 þ4ðkn rho Þ ð1h b23 þ4ðkn rho Þ 0:5 b3 h c e e
By using the same procedure, temperature distribution for the plate 2 is readily found to be
ð8aÞ
0:5 b3 þ
hh
D¼
F nf ¼
It is noted that separation constant, k is determined afterward from temperature boundary conditions. Eqs. (7a) and (7b) has analytical solution as the follow, respectively
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Þ b3 þ b23 þ4ðkn rho Þ ð1h c
ð12aÞ
ð12bÞ
kn
The left-hand side of Eq. (6) is a function of the variable r only and the right-hand side is a function of z only. Therefore, it is conclude that the only way that the above equation can hold is
J 0 ðknr i Þ Y 0 ðknr Þ Y 0 ðknr i Þ
C 0 ðknr Þ
ð15Þ
By substituting of Eqs. (1) and (4) into Eq. (2b) and applying separation of variables method, two ordinary differential equations in terms of only radial and thickness variables for annular face sheets can be derived
@2 1 @ Rðr Þ þ RðrÞ þ k2 Rðr Þ ¼ 0 r @r @r 2 ki
2 @2 h zÞ k2 Zð ZðzÞ ¼ 0 ði ¼ m; cÞ b @ z2
ð16aÞ
ð16bÞ
Temperature distribution for metal layer is computed by using m Þ ¼ T mo ; and one non-homogeneous boundary condition, h1 ðr; h applying the orthogonality property
Tm ¼
1 h kn X h kn r o km z Em ero km z C 0 ðknr Þ n e
ð17Þ
n¼1
where
Em n ¼
T m0 h kn r o k m hm
e
h kn
ero km hm
D
As before, temperature distribution for ceramic layer due to the one non-homogeneous boundary condition, h1 ðr; 1Þ ¼ T h can be found as
C 0 ðknrÞ
ð11Þ
hc1 ¼
1 h kn X h kn h kn Ecn ero kc z e2ro kc ð1hc Þ ero kc z C 0 ðknr Þ n¼1
ð18Þ
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where
Ecn
¼
T h h kn
h kn
ero kc ero kc
D
1 h kn X h kn hkn F cn ero kc z e2ro kc hc ebkc z C 0 ðknr Þ
r z;z ¼ b1 r z z;z ¼ u
here
T ci h kn ro kc ð1hc Þ
e
h kn
h kn
e2ro kc ero km ð1hc Þ
D
ð20Þ
Temperature distribution for ceramic layer is determined by applying superposition method to Eqs. (18) and (19) 1 h kn X h kn h kn h kn Tc ¼ Ecn Ecn þ F cn ero kc z Ecn e2ro kc ð1hc Þ þ F cn e2ro kc ero kc z C 0 ðknr Þ
2 E0 h m h 1 1 r z;r þ u u u r r; r r; r r r ro 1 m ro 1 m2 r 2 0 b ðzhm Þ h E0 a 1 s rz þ he 2 T b ro 1 m
In this section thermo-elasticity of a sandwich circular plate with arbitrary edges boundary conditions is studied. Stress distributions in the FGM core layer can be written as
r r ¼
r h ¼
r s
¼ ðri ; srz Þe
b1 ðzhm Þ
ði ¼ r; h; zÞ
ð22Þ
From Eqs. (1), (4) and (22), non-dimensional thermo-elastic constitutive relations in term of displacements are introduced as
r r ¼
0 E h h r;r þ mu r þ mu z;z ð1 mÞ u ro ro ð1 þ mÞð1 2mÞ 0 b ðzhm Þ E0 a T e2 1 2m
srz ¼
0 E h r;z z;r þ u u 2ð1 þ mÞ ro
ð23Þ
Dimensionless governing equilibrium equation, in the absence of body force, for axisymmetric problems of FGM layer is
h h 1 r r;r þ srz;z þ ðr r r h Þ þ b1 srz ¼ 0 ro ro r
z ¼ 0 u
ð27bÞ
ð1 mÞ
r z þ
Since state Eq. (26) is not possible to solve analytically so, a semi-analytical procedure with the aids of DQ technique is use to solve it. In DQ method, the nth-order partial derivative of a continuous function f(x, z) with respect to x at a given point xi can be approximated to a linear sum of weighted function values at all of the discrete points as
ð28Þ
zi;z u
ð24Þ
ð25aÞ
!
N X ð1 þ mÞð1 2mÞ h m 1 ri þ rj ¼ r zi g ij u u ro 1 m ri E0 ð1 mÞ j¼1
þ
r;z ¼ u
And the pertinent relations for Simply (S) support, Clamped (C) support, Free (F) from supported at the r ¼ r i and r ¼ 1 edges are as;
r ¼ 0; S:r
0 m 1 0 b ðzhm Þ E0 a h E T þ m e2 u u r r; r r o 1 m2 r 1m
m
N X h 1 rzi þ rzj r zi;z ¼ b1 r zi s g ij s r o ri j¼1
h h 1 srz;r þ r z;z þ srz þ b1 r z ¼ 0 ro ro r
ð27aÞ
where N is the total number of discretization sampling points, and g nij are the xi -dependent weighted coefficients [29]. Applying Eq. (28) to Eq. (26) results in the following state equations at an arbitrary sampling point ri
0 E h h m ur;r þ m ur þ ð1 mÞuz;z ro ð1 þ mÞð1 2mÞ r o 0 b ðzhm Þ E0 a e2 T 1 2m
m E a E h r z þ z 0 2 h ur þ ur;r 0 0 eb2 ðzhm Þ T ro 1 m r ð1 mÞ 1m
m
n N X @f ðxi ; zÞ ¼ g nij f ðxj ; zÞ ði ¼ 1; . . . ; N n ¼ 1; . . . ; N 1Þ n @x x¼xi j¼1
0 h h E r h ¼ m ur;r þ ð1 mÞur þ muz;z ro ð1 þ mÞð1 2mÞ r o 0 b ðzhm Þ E0 a T e2 1 2m
r z ¼
ð26Þ
In-plane normal stresses in term of state variables are found from Eqs. (23) and (26) as the follow
2.2. Thermo-elastic problem for FGM layer
z 2ð1 þ mÞ h @u srz þ 0 r o @r E
srz;z ¼
ð21Þ
0 rz
h 1 rz;r rz þ s s r o r
ð1 þ mÞð1 2mÞ m h 1 @ z u r þ u r r r; 0 ð1 mÞ @r 1 m ro r E ð1 þ mÞa0 b2 ðzhm Þ þ e T 1m
r;z ¼ u
n¼1
0 i;
ð25cÞ
Using Eqs. (23) and (24) leads to the following state-space equations
ð19Þ
n¼1
F cn ¼
ð25bÞ
rz ¼ 0 r ¼ s F:r
By using the same procedure for the ceramic with the one nonc h Þ ¼ T ci ; temperhomogeneous boundary condition, h2 ðr ; 1 h f ature distribution will be
hc2 ¼
z @u ¼0 @r
r ¼ u z ¼ 0; C:u
!
0 b ðzhm Þ ð1 þ mÞa Ti e2 1m
N hX 2ð1 þ mÞ zj þ g u srzi 0 r o j¼1 ij E
2 N E0 h m X h zj þ g ij r r o 1 m j¼1 r o 1 m2 ! N N 0 a X 0 b ðzhm Þ ri 1 X h E u 2 1 s rj rj b rzi þ he 2 Ti g ij u g ij u r r 2i r 1 m o j¼1 j¼1
srzi;z ¼
ð29Þ
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Similarly, from Eqs. (27a,b) and (28) it can be written
r ri ¼
m ð1 mÞ
r zi þ
N 0 h E m X rj g ij u uri þ 2 ro 1 m ri j¼1
!
Table 1 3 The deflection factor a ¼ wð0; 0:5Þ hb of simply supported FGM plate.
0 a 0 b ðzhm Þ E Ti e2 1m ð30aÞ
r hi ¼
m ð1 mÞ
r zi þ
N 0 m X h E 1 ri þ m g ij u rj u 2 ro 1 m ri j¼1
!
0 a 0 b ðzhm Þ E T i e2 1m ð30bÞ
where rki ¼ rk ðri ; zÞ ðk ¼ r; h; zÞ szri ¼ szr ðri ; zÞ uki ¼ uk ðri ; zÞ ðk ¼ r; zÞ. Assembly of Eq. (29) at all sampling points leads to the following global state equation in the matrix form
d df ¼ Gf df þ Pf T f dz
ð31Þ
where
z df ¼ f r
z u
r u
srz gT ; r z ¼ ½r z1 ; r z2 ; . . . ; r zN T
ð32Þ
Coefficient partitioned matrix Gf ; Pf and T f are described in Appendix A, and other sub-vectors in Eq. (31) are defined in the same manner as Eq. (32). Applying the boundary conditions at r ¼ ri ; 1 to the Eq. (31) results in the unique solution for the state variables, d. Taking into account the boundary conditions, the state-space Eq. (31) is converted to the following equation which has unique solution
d dfb ¼ Gfb dfb þ Pfb T f dz
ð33Þ
where the subscript, b, denotes that the state equation contains the boundary conditions and the matrix Gfb and Pfb according to each boundary condition type are defined in Appendix A.
k1
5 4 3 2 1 0 1 2 3 4 5
h b
¼ 0:1
h b
¼ 0:2
h b
¼ 0:3
Present
[11]
Present
[11]
Present
[11]
91.944 56.863 34.297 20.410 12.143 7.300 4.456 2.745 1.692 1.028 0.610
91.947 56.866 34.300 20.412 12.144 7.301 4.457 2.747 1.694 1.031 0.613
47.107 29.291 17.753 10.601 6.312 3.784 2.294 1.401 0.854 0.512 0.301
47.110 29.295 17.757 10.604 6.314 3.786 2.296 1.403 0.857 0.516 0.304
32.636 20.466 12.500 7.503 4.473 2.670 1.603 0.967 0.580 0.342 0.196
32.640 20.470 12.504 7.507 4.476 2.673 1.606 0.969 0.582 0.345 0.200
Analytical solution to Eq. (33) is
Z Þþ dfb ðzÞ ¼ eGfb z eGfb hm dfb ðh m
z
m h
m 6 z 6 1 h c h
eGfb g Pfb T f ðgÞdg
at ð34aÞ
2.3. Thermo-elastic problem of face sheets Governing state space differential equations containing edges boundary conditions for face sheets are derived from Eq. (33) by canceling b1 which their solution for metal and ceramic facing sheets are respectively as the follow
Z z at dmb ðzÞ ¼ eGmb z dmb ð0Þ þ eGmb g Pmb T m ðgÞdg
a. Radial normal stress
m 0 6 z 6 h
0
ð34bÞ
b. Circumferential normal stress
c. Transverse displacement Fig. 2. Convergence of the proposed method for sandwich annular plate with FGM core and S–S boundary condition.
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
" Gcb z
dcb ðzÞ ¼ e
e
m þh Þ Gcb ðh f
dcb h m þ hf þ
Z
z
þh h m f
e
Gcb g
# Pcb T c ðgÞdg
6 z 6 1 þh h m f
at
ð34cÞ
From Eqs. (28) and (30), in-plane stresses of face sheets in term of state variables are
E h m r ri ¼ r zi þ z j 2 h j uri þ r o 1 mj ð1 mj Þ r
mj
N X
!
rk g ik u
k¼1
j a j E T 1 mj ð35aÞ
r hi ¼
mj ð1 mj Þ
r zi þ
N j mj X h E 1 rj ri þ mj g ij u u 2 r o 1 mj r j¼1
!
j a j E T 1 mj
r z ¼ srz ¼ 0 at z ¼ 0
c. Transverse normal stress
e. Transverse displacement
p Þ ¼ M m ðdm ð0Þ þ Im Þ dm ðh
ð37aÞ
m þ h Þ ¼ M d ðh m Þ þ B I df ðh f f f f f
ð37bÞ
m þ h Þ þ Bc Ic dc ð1Þ ¼ Mc dc ðh f
ð37cÞ
Mm ¼ eGm hm ; Z
m h
Im ¼
M f ¼ eGf hf ;
Z Ic ¼
eGmb g :Pmb T m ðgÞdg;
1
m þh h f
Z If ¼
Bf ¼ eGf ðhm þhf Þ ;
M c ¼ e Gc hc ;
0
ð36aÞ ð36bÞ
a. Radial normal stress
Relation between state variable at the top and bottom surfaces of metal, FGM and ceramic layers are obtained from Eqs. (34a)–(34c) respectively as follow
where
ð35bÞ
where j ¼ c; m. Surface traction boundary condition at the top surface of the ceramic and bottom surface of the metal layers are, respectively
r z ¼ p0 ; srz ¼ 0 at z ¼ 1
2.4. Global transfer matrix and surface boundary conditions
m þh h f
m h
eGfb g :Pfb T f ðgÞdg
eGcb g Pcb T c ðgÞdg
b. Circumferential normal
Bc ¼ eGc
stress
d. Radial displacement
f. Temperature
Fig. 3. Effect of different lay up of sandwich plate on thermomechanical properties of sandwich circular plate with C–S boundary conditions.
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
Since the state variables at the interfaces of the metal, FGM and ceramic layers are continuous, so from Eqs. (37a)–(37c), following relation between the state variables at the top surface of ceramic and the bottom surface of metal is found
dc ð1Þ ¼ M dm ð0Þ þ B
ð38Þ
where N 1 ¼ Mc IF M f Mm and N2 ¼ Mc IF Mf M m Im þ Mc IF Bf If þ Bc Ic are the global transfer matrices. Imposing surface traction at the top and bottom surfaces of ceramic and metal layers respectively, Eqs. (36a) and (36b), to the Eq. (38) results in the following equations
8 9 P 0 > > > > > > > > < uz = > > > > :
2
M 11
6 6 M 21 ¼6 6 > ur > 4 M 31 > > ; 0 M 41 h
M 12
M 13
M 22
M 23
M 32
M 33
M 32
M 33
8 9 38 9 M14 > 0 > B1 > > > > > > > > > > > > > 7> < < = M24 7 uz B2 = 7 þ 7 > M34 5> B3 > > > ur > > > > > > > > > : > : > ; ; M34 0 0 B4
d. Transverse shear stress
uz ur
¼ 0
M 12 M 42
1
P0 0
B1 B4
ð40Þ
3. Numerical results and discussion In present work, sampling points with the following coordinates [29] are used;
ð39Þ
M 13 M 43
Then by using the obtained state variable at the bottom surface of the metal and using Eqs. (34a)–(34c), the state variables along the thickness of sandwich plate are determined. In-plane stresses are then computed by using Eqs. (35a) and (35b).
xj ¼
where Mij and Bi are square and column sub-matrixes with dimension depending on the case of edge boundary conditions. By solving Eq. (39), displacements at the bottom surface of metal layer are computed as follow
a. Radial normal stress
1 cos
ðj 1Þp r o r i N1 2
j ¼ 1; 2; . . . ; N
The material properties of FGM core layer is made by Metal/ Ceramic with the following material properties
Em ¼ 227:24 GPa; am ¼ 15 106 =K; km ¼ 25 W=mK for Metal Ec ¼ 125:83 GPa; ac ¼ 10 106 =K; kc ¼ 2:09 W=mK for Ceramic
b. Circumferential
normal stress
e. Radial displacement
f. Transverse displacement
Fig. 4. Effect of edges boundary conditions on the through-thickness distribution of displacements and stresses at the mid radius, r ¼ ro 2þri for hybrid annular FGM plate.
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
a. Radial normal stress
b. Circumferential normal stress
c. Transverse normal stress
d. Transverse shear stress
e. Radial displacement
f. Transverse displacement
g. Temperature Fig. 5. Influence of thermal load on the stresses, displacements distributions across the thickness of FGM annular plate with c–c condition.
m ¼ 0:3; T h ¼ 350 K;
hFG ¼5 hm
To show convergence of the present formulation, numerical results of dimensionless deflection as well as radial and circumferential normal stresses are computed and plotted in Fig. 2. As the figures show non-dimensional radial stresses is independent of increasing sampling points. Also it is observed that distribution of dimensionless deflection and circumferential stress along the thickness direction at mid radius of sandwich plate approaches
to a specific constant value at about 14 discrete points which confirms good convergence of the method. Since thermoelastic response of circular sandwich plate has not yet been reported in the literature, so in order to validate the present approach, a comparison study for the thermoelastic behavior of circular plate without face sheets is carried out. Numerical result of deflection factor (a) for circular FGM plate subjected to mechanical load is computed and along with the reported results by Yun et al. [11] are tabulated in Table 1. Regardless of the amount of thickness-toradius ratio and gradient index, deflection factor of present
A. Alibeigloo / Composite Structures 136 (2016) 229–240
formulation is smaller than that of Ref. [11]. According to the table in isotropic material, k1 ¼ 0, the results of the present approach is exactly the same as the results obtained by Yun et al. [11]. Furthermore it can be observed that by increasing the gradient index as well as the thickness to radius ratio, difference of two results increases. It is noted that the discrepancy of the two results is due to the division of FGM layer to the some fictitious isotropic layers with different thermoelastic constants in Ref. [11]. Effects of lay-up on thermo-elastic behavior of the sandwich annular plate subjected to thermal load and with S–C boundary conditions are considered in Fig. 3a–f. According to the figures in the sequence of metal, FGM and ceramic layup, M/FGM/C, through the thickness distribution of stresses, displacement and temperature at a given point are always smaller in magnitude than those at the corresponding points in the sequence of ceramic, FGM and metal, C/FGM/M, layup. In addition, From Fig. 3a and b it is concluded that the interface in-plane stresses are continuous. Through thickness distribution of stresses and displacements of sandwich annular plate with various edges condition are presented in Fig. 4a–e. From Fig. 4a–c it is seen that all of stresses distribution in S–S boundary condition, compared with the other edges conditions, has minimum value and in C–C boundary conditions has maximum value. Moreover, it can be concluded that stresses distribution in FGM layer is nonlinear function of transverse coordinate whiles it is linear in facing sheets and this is due to the variation of material properties in FGM layer according to the exponential function of transverse coordinate. Fig. 4d depicts the effect of edges boundary conditions on through the thickness distribution of radial
a. Radial normal stress
c.Transverse normal stresst
e. Radial displacement
237
displacements. According to the figure this distribution for the plate with C–C boundary conditions is minimum whereas it is maximum for the plate with S–S boundary condition. As the Fig. 4e shows, distribution of transverse displacement along the thickness direction is maximum for C–F boundary condition whereas it is minimum value for C–C boundary condition. Fig. 5a–g depicts through-thickness distributions of the dimensionless stresses, displacement and temperature with various surface temperature difference conditions. As the Fig. 5a and b show dimensionless radial and circumferential stresses increases by increasing surface temperature difference whiles transverse normal stress (Fig. 5c) decrease. Besides, it is concluded that distribution of radial and circumferential normal stresses at top surface is more affected by temperature difference than that at the bottom surface whereas this is converse for the transverse normal stress rz ; (Fig. 5d) increase by distribution. Transverse shear stress, s increasing the temperature difference in the region nearly bellow the mid surface and then decreases above the mid surface of the sandwich plate. From Fig. 5e and f it can be observed that radial displacement increases and transverse displacement decreases when surface temperature difference increases. Through the thickness temperature distribution is represented in Fig. 5g. According to the figure and as expected, increase surface temperature difference causes to increase temperature gradient along the thickness direction. Influence of surface tractions on through the thickness distribution of displacements and stresses for sandwich circular plate with FGM core is depicted in Fig. 6a–f. As the Fig. 6a shows, the neutral surface does not coincide with the mid
b. Circumferential normal stress
d. Transverse shear stress
f. Transverse displacement
Fig. 6. Influence of mechanical load on the stresses, displacements distributions across the thickness of FGM annular plate with c–c condition.
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Table 2 Non dimensional thermal stresses sandwich circular plate with S–S, P = 0, T 1 ¼ 300 K T 2 ¼ 350 K, r ¼ 0:5ðr o þ r i Þ, hhFG ¼ 5. m h
h
h
h
ro ri
ro h
r r
2
10 20 30 40 50 60 70
50.7345 61.3998 64.6837 66.3456 67.3640 68.0269 68.4583
0.4575 0.3792 0.1783 0.0638 0.0215 0.0064 0.0013
92.0854 13.2075 50.9592 70.9935 83.5109 91.9009 97.6917
11.5524 6.9198 6.0386 4.9821 4.0491 3.3167 2.7615
3
10 20 30 40 50 60 70
49.3851 64.7846 70.3959 73.3464 75.0967 76.1712 76.8388
0.3397 0.3017 0.0738 0.0114 0.0038 0.0064 0.0058
27.7063 48.2287 75.7091 90.1713 98.7314 103.9813 107.2398
14.6671 9.6147 7.1723 5.4440 4.2709 3.4691 2.9060
10 20 30 40 50 60 70
47.3694 64.7281 70.7991 73.8992 75.6593 76.6809 77.2741
0.3934 0.2253 0.0390 0.0019 0.0090 0.0085 0.0067
3.5841 60.2734 83.3312 95.2430 102.0131 105.9569 108.2723
17.3542 10.9166 7.7273 5.7359 4.4685 3.6302 3.0499
4
2
r z
2
r h
2
srz
2
thickness of the FGM core and it is shifted toward the stiffer surface. Moreover it is seen that stresses distribution satisfies surfaces boundary condition. According to the figures and as expected increase mechanical load causes to increase stresses as well as displacements. Effect of outer radius to thickness ratio, rho , on thermal stresses at mid radius and mid surface of plate with S-S boundary conditions for various outer radius to inner radius ratio, rro ; are i
computed numerically and tabulated in Table 2. It is shown in Table 2 that increase rho causes to increase radial and circumferential stresses and to decrease the transverse normal and shear stresses. In addition, it is observed that when rroi increases, all of stresses except to transverse shear stress decreases. 4. Conclusion In this investigation axisymmetric thermoelastic analysis of sandwich circular plate with FGM core imbedded in metal and ceramic facing sheets with various boundary conditions is carried out using semi-analytical state space differential quadrature method. Analytical solution along the thickness direction is performed by using state space technique and approximate solution in the radial direction is obtained by applying one-dimensional differential quadrature method. This method overcomes the difficulty conformed by the traditional state space method when applied to analyses the plate with non-simply supported boundary conditions. From parametric study it was concluded that in the sequence of metal, FGM and ceramic layup, M/FGM/C, through the thickness distribution of stresses, displacement and temperature at a given point are always smaller in magnitude than those at the corresponding points in the sequence of ceramic, FGM and metal, C/FGM/M, layup. Also numerical results predicted the continuity of interface in-plane stresses. The neutral surface of the FGM is not at the mid-surface but depends on the through-thickness variation of Young’s modulus. Stress field in S–S boundary condition has minimum value and in C–C boundary conditions it is maximum value. Increasing the surface temperature difference causes to increase in-plane normal stresses and to decrease transverse normal stress. It was concluded that radial displacement increases and transverse displacement decreases when surface temperature difference increases.
Appendix A 3 1 IN 0N 0N rho g ij þ 1r ij b 7 6 7 6 1 7 6 ð1þmÞð12mÞ h m 0N 0N ro 1m g ij þ r ij 7 6 E0 ð1mÞ IN 7 6 Gf ¼ 6 7 2ð1þmÞ 7 6 h 0N ro g ij 0N 7 6 E 0 7 6 5 4 2 2 E0 h 2 1 1 b rho 1m m g ij 0N g g þ I 1 N ij ro r ij ij r ij 1m2 2
ði; j ¼ 1; . . . ; NÞ
1
where
r ij
¼
1 ri
0
i ¼ j i–j
i; j ¼ 1; . . . ; N
2 3 T1 6 7 6 T2 7 6 7 6 7 6 : 7 7 T f ¼ 6 6 : 7 6 7 6 7 6 : 7 4 5 T N
3
2
0N 7 6 6 ð1 þ mÞIN 7 a 0 b2 ðzhm Þ 7 6 Pf ¼ 6 e 7 7 1m 6 0 N 5 4 0 h g T E r o ij S–S –FGM layer
2
1 IðN2Þ b 6 6 ð1þmÞð12mÞ 6 E0 ð1mÞ IðN2Þ 6 6 0NðN2Þ Gfb ¼ 6 6 6 4 rho 1m m g Tss
0ðN2Þ 0ðN2Þ
0ðN2ÞN rho
rho g Tss
m
1m
2 h ro
0NðN2Þ
rho ðg ss þ ass Þ
ðg ss þ ass Þ
0ðN2ÞN
0N
2ð1þmÞ IN E
E 0 1m2
0
ðkss Þ
1 IN b
ði; j ¼ 1; . . . ; NÞ 2
3
0ðN2Þ
7 6 6 ð1 þ mÞIðN2Þ 7 a 0 b2 ðzhm Þ 7 6 Pfb ¼ 6 e ; 7 6 0NðN2Þ 7 1 m 5 4 0 h g T E r o ss
g ssij ¼ g ij i ¼ 2; . . . ;
N 1 J ¼ 1; . . . ; N where
1 1 ass ¼ 0ðN2Þ1 0ðN2Þ1 r ss r ssij (1 i¼j ri i; j ¼ 2; . . . ; N 1 ¼ 0 i–j kss ¼ g 2ss þ f ss þ f ssn þ ¼ 1; . . . ; N;
2 1 þ g ss2 þ bss ; r ss2
f ssij ¼ g i1 g 1j i; j ¼ 1; . . . ; N
f ssnij ¼ g iN g Nj i; j ¼ 1; . . . ; N;
¼
8 2 < 1 :
ri
mg i1 r1
i¼j
2 1 r ss2ij
i; j ¼ 1; . . . ; N
i–j
0
bssij ¼
g 2ssij ¼ g 2ij i; j
0NðN2Þ N1
C–C –FGM layer
mg iN rN
; N1
g ss2ij ¼
g ij i; j ¼ 1; . . . ; N ri
3 7 7 7 7 7 7 7 7 5
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A. Alibeigloo / Composite Structures 136 (2016) 229–240
2
1 IðN2Þ b 6 6 ð1þmÞð12mÞ 6 6 E0 ð1mÞ IðN2Þ 6 6 0ðN2Þ Gfb ¼ 6 6 6 4 rh0 1m m g cc
rh0 ðg cc þ ð1r Þcc Þ
0ðN2Þ
0ðN2Þ h m r0 1m g cc þ 1r cc
rh0 g cc
0ðN2Þ
2ð1þmÞ IðN2Þ E
0ðN2Þ
2 h r0
0ðN2Þ
E 0 1m2
g cc2 g 2cc þ
0ðN2Þ 0
1 2
1 IðN2Þ b
r cc
3
2
7 7 7 7 7 7 7 7 7 5
6 6 6 6 6 6 Gfb ¼ 6 6 6 4
ði;j ¼ 1;::; NÞ
g 2ccij ¼ g 2ij i; j ¼ 2; . . . ; N 1
m
ðg g þ g iN g Nj Þ i; j ¼ 2; . . . ; N 1; 1 2m i1 1j 8 2 < 1 i¼j ri i; j ¼ 2; . . . ; N 1 ¼ : 0 i–j
g 0cfij
2 1 r ccij
¼ g ij i; j ¼ 2; . . . ; N 1;
acf ¼
kcf ¼ g cf 2 g 2cf þ f cf þ ccf þ dcf ;
0
1 IðN1Þ b
#
0ðn2Þðn1Þ 01ðn1Þ
g cf 2ij ¼
g 2cfij ¼ g 2ij i ¼ 1; . . . ; N 1 J ¼ 2; . . . ; N; "
0ðN2Þ
0ðN2ÞðN1Þ
rho ðg cs þ acs Þ 0ðN2ÞðN1Þ 2ð1þmÞ IðN1Þ E 0
1 IðN1Þ b
3 7 7 7 7 7 7 7 7 5
ði; j ¼ 1; . . . ; NÞ 3 0ðN2Þ 7 6 6 ð1 þ mÞIðN2Þ 7 a 0 b2 ðzhm Þ 7 6 Pfb ¼ 6 e ; 7 6 0ðN1ÞðN2Þ 7 1 m 5 4 0 h g T E r o cs 2
g csij ¼ g ij i ¼ 2; . . . ; N 1 J ¼ 2; . . . ; N g 2csij ¼ g 2ij i; j ¼ 2; . . . ; N; (1
i¼j 1 1 r i; j ¼ 2; ::; N 1 0ðN2Þ1 ; ¼ i acs ¼ r cs r csij 0 i–j i¼j i–j
g ij i; j ¼ 1; . . . ; N; ri
i; j ¼ 2; . . . ; N;
f csij ¼
m2 1 2m
1 g cs2ij f csij r cs2ij
C–F –FGM layer
0ðN1Þ
0ðN1Þ 2ð1þmÞ IðN1Þ E
7 7 7 7 7 7 7 7 7 5
1 rN
g ij i ¼ 1; . . . ; N 1 J ri
f cfij ¼ g iN g Nj i
¼ 1; . . . ; N 1 J ¼ 2; . . . ; N
1 IðN2Þ b
kcsij ¼ g 2csij þ
g cf
0ðN1Þ 2 E0 h ðkcf Þ ro 1m2
r cf
¼ 2; . . . ; N
6 6 ð1þmÞð12mÞ I 0ðN2Þ rho 1m m ðg cs þ acs Þ 6 E0 ð1mÞ ðN2Þ 6 6 6 rho g Tcs 0ðN1Þ Gfb ¼ 6 0ðN1ÞðN2Þ 6 4 2 E0 h rho 1m m g Tcs 0ðN1ÞðN2Þ ðkcs Þ ro 1m2
g ss2ij ¼
m
1m
þ hcf
rho g cf þ
( 1 i¼j 1 ri i; j ¼ 1; . . . ; N 1 g cfij ¼ g ij i; j ¼ 1; . . . ; N 1; ¼ r cf 0 i–j
C–S –FGM layer
8 < 1 2 1 ri ¼ r cs2 : 0
rho
rho g 0cf
g0 1m cf m
"
2
2
0ðN1Þ
rho
3 0ðN1ÞðN2Þ 7 6 ð1 þ mÞI ðN2Þ 7 6 7 6 7 a0 eb2 ðzhm Þ 0 Pfb ¼ 6 1ðN2Þ 7 1m 6 7 6 0 ðN1ÞðN2Þ 5 4 0 h g E r o cf 1
g ij g cc2ij ¼ i; j ¼ 2; . . . ; N 1; ri ( 1 i¼j 1 ri i; j ¼ 2; . . . ; N 1 ¼ r ccij 0 i–j f ccij ¼
0ðN1Þ
0ðN1Þ
2
0ðN2Þ 7 6 6 ð1 þ mÞIðN2Þ 7 a 0 b2 ðzhm Þ 7 6 Pfb ¼ 6 e ; 7 7 6 1 m 0 ðN2Þ 5 4 0 h g E ro cc g ccij ¼ g ij i; j ¼ 2; . . . ; N 1;
acf
0ðN1Þ
1 3
ði; j ¼ 1; . . . ; NÞ
3
2
1 IðN1Þ b
g cs2ij ¼
g ij i; j ¼ 2; . . . ; N ri
g i1 g 1j g i1 g 1j i; j ¼ 2; . . . ; N
ccf ¼
01ðN2Þ 1 2 r cf
g cf 1ij ¼ g ij
# 0ðN2Þ1
mg jN ; dcf ¼ 0ðn1ÞðN2Þ ; j ¼ 1; . . . ; N 1 rN
i ¼ 1; . . . ; N 1; j ¼ 2; . . . ; N 1:
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