Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings
L.S.Ma,T.J.Wang *
Department of Engineering Mechanics,Xi’an Jiaotong University,Xi’an 710049,China
Received 21August 2002;received in revid form 31January 2003
Abstract
Bad on the classical nonlinear von Karman plate theory,axisymmetric large deflection bending of a functionally graded circular plate is investigated under mechanical,thermal and combined thermal–mechanical loadings,respec-tively,and axisymmetric thermal post-buckling behavior of a functionally graded circular plate is also investigated.The mechanical and thermal properties of functionally graded material (FGM)are assumed to vary continuously through the thickness of the plate,and obey a simple power law of the volume fraction of the constituents.Governing equations for the problem are derived,and then a shooting method is employed to numerically solve the equations.Effects of material constant n and boundary conditions on the temperature distribution,nonlinear bending,critical buckling temperature and thermal post-buckling behavior of the FGM plate are discusd in details.
Ó2003Elvier Science Ltd.All rights rerved.
Keywords:FGM;Nonlinear bending;Buckling;Post-buckling;Thermal and mechanical loading;Circular plate;Shooting method
1.Introduction
Extensive investigations on the thermal bending and post-buckling of isotropic and composite plates and shells were carried out by Tauchert and Huang (1987);Tauchert (1991);Meyers and Hyer (1991)and Leissa (1992),etc.However,there are few works on the stability,vibration,bending and buckling behavior of functionally graded structures,and the are still open problems.Loy et al.(1999)and Pradhan et al.(2000)examined the free vibration of functionally graded cylindrical shell by using the Rayleigh–Ritz method.An analytical solution of the dynamic respon of simply supported functionally graded cylinder due to low-velocity impact was given by Gong et al.(1999).Bad on the classical small deflection theory of plate,Yang and Shen (2001)investigated the dynamic respon of a functionally graded rectangular thin plate with initial stress subjected to partially distributed impulsive lateral loads and without or resting on an *Corresponding author.Tel.:+86-29-2668757;fax:+86-29-3237910.
E-mail address:wangtj@mail. (T.J.Wang).
0020-7683/03/$-e front matter Ó2003Elvier Science Ltd.All rights rerved.
doi:10.1016/S0020-7683(03)00118-5
3312L.S.Ma,T.J.Wang/International Journal of Solids and Structures40(2003)3311–3330
foundation.Ng et al.(2000,2001)studied the dynamic stability of functionally graded rectangular plate and cylindrical shell,respectively.A modified classical lamination theory to account for piezoelectric coupling terms under applied electricfield was developed by Almajid et al.(2001),and then the theory was applied to predict the out-of-plane displacement and stressfield of actuators and the functionally graded material(FGM)bimorph.Mian and Spencer(1998)established a large class of exact solutions of the three-dimensional elasticity equations for functionally graded and laminated elastic materials.
The respon of functionally graded ceramic–metal plate accounting for the transver shear strains, rotary inertia and moderately large rotations in the von Karman n was studied by Praveen and Reddy (1998),in whichfinite element method was employed to investigate the static and dynamic respons of the functionally graded plate by varying the volume fraction of the ceramic and metallic constituents.Effect of impod temperaturefield on the respon of the functionally graded plate was also discusd.Reddy and Chin(1998)investigated the dynamic thermo-elastic res
pon of functionally graded cylinders and plates.A thermo-elastic boundary value problem was derived by using thefirst-order shear deformation plate theory that account for coupling with a three-dimensional heat conduction equation for a functionally graded plate.Using thefirst-order shear deformation theory of Mindlin plate,axisymmetric bending of func-tionally graded annular and circular plates was studied by Reddy et al.(1999),in which the solutions were expresd in terms of the classical solutions bad on the Kirchhoffplate theory.Bad on the higher-order shear deformation theory of plate,Reddy(2000)developed both theoretical andfinite element formulations for thick FGM plates,and the nonlinear dynamic respons of FGM plates subjected to suddenly applied uniform pressure were studied.Bad on the von Karman theory,Woo and Meguid(2001)derived an analytical solution expresd in terms of Fourier ries for the large deflection of functionally graded plates and shallow shells under transver mechanical loading and a temperaturefield.Cheng and Batra(2000) studied three-dimensional thermo-mechanical deformations of an isotropic linear thermo-elastic func-tionally graded elliptic plate.A clod form solution was obtained which shows that the through-thickness distribution of the in-plane displacements and transver shear stress in a functionally graded plate do not agree with tho assumed in classical and shear deformation plate theories.Moreover,a new t offield equations in terms of displacement and stress potential functions for inhomogeneous plates had been prented and reformulated by Cheng(2001),and mixed Fourier
ries technique was employed to solve the equations.Using an asymptotic method,the three-dimensional thermo-mechanical deformations of func-tionally graded rectangular plate were investigated by Reddy and Cheng(2001)and the distributions of temperature,displacements and stress in the plate were calculated for different volume fraction of ceramic constituent.
Assuming that the material properties throughout the structure are produced by a spatial distribution of the local reinforcement volume fraction v f¼v fðx;y;zÞ,Feldman and Aboudi(1997)studied the elastic bifurcation buckling of functionally graded plate under in-plane compressive loading.More recently, Javaheri and Eslami(2002a,b)studied the thermal buckling of functionally graded rectangular plate bad on the classical and the higher-order shear deformation theories of plate,respectively,and obtained the clod form solutions under veral types of thermal loads.Ma and Wang(in press)studied the axisym-metric post-buckling behavior of a functionally graded circular plate under uniformly distributed radial compression on the basis of classical nonlinear plate theory.
To the authorsÕknowledge,only few works on the nonlinear bending of functionally graded plates are concerned,but the thermal post-buckling of FGM plates has not been carried out in the previous works.In the prent paper,axisymmetric nonlinear bending and thermal post-buckling behavior of a functionally graded circular plate are studied under mechanical,thermal and combining thermal–mec
hanical loading in the framework of von Karman plate theory.Simply supported and clamped boundary conditions are considered.The material properties are assumed to vary continuously through the thickness of the plate. Effects of material properties and boundary conditions on the large deflection bending and thermal post-buckling behavior of the FGM plate are discusd in details.
2.Basic equations
A functionally graded circular plate with thickness h and radius b is considered here.It is assumed that the mechanical and thermal properties of FGM vary through the thickness of plate,and the material properties P can be expresd as (Reddy and Chin,1998;Reddy et al.,1999)
P ðz Þ¼ðP m ÀP c ÞV m þP c ;ð1Þwhere the subscripts m and c denote the metallic and ceramic constituents,respectively,V m denotes the volume fraction of metal and follows a simple power law as V m ¼h À2z 2h
n ;ð2Þwhere z is the thickness coordinate (Àh =26z 6h =2),and n is a material constant.According to this dis-tribution,bottom surface (z ¼Àh =2)of the functionally graded plate is pure metal,the top surface (z ¼h =2)is pure ceramics,and for different values of n one can obtain different volume fracti
ons of metal.
Bad on the classical nonlinear von Karman plate theory,the equilibrium equations of a thin plate subjected to a thermal load T and uniformly distributed transver mechanical load q are as follows
ðrN r Þ;r ÀN h ¼0;
ð3ÞðrQ r Þ;r þðrN r W ;r Þ;r ¼Àrq ;
ð4ÞðrM r Þ;r ÀM h ÀrQ r ¼0;ð5Þwhere the comma followed by r denotes differentiation with respect to r ,W is the displacement in z di-rection,and the force and moment components N and M are as follows,
ðN r ;N h Þ¼
Z h =2
Àh =2
ðr r ;r h Þd z ;ð6a ÞðM r ;M h Þ¼Z h =2
Àh =2ðr r ;r h Þz d z :
ð6b Þ
The constitutive relations for the FGMs are given by
r r r h ¼E ðz Þ1Àm 21m m 1 e 0r e 0h þz j r j h ÀE ðz Þ1Àm a ðz ÞT ðz Þ11 :ð7Þ
The Young Õs modulus E ðz Þand thermal expansion coefficient a ðz Þin Eqs.(7)follow the distribution law of Eqs.(1)and (2),namely,
E ðz Þ¼ðE m ÀE c ÞV m þE c ;
a ðz Þ¼ða m Àa c ÞV m þa c :
For simplicity,the Poisson Õs ratio m in Eqs.(7)is assumed to be a constant.The radial and circumferential
strain components e 0r and e 0h in the mid-plane of the plate (i.e.z ¼0)can be calculated as
e 0r ¼d U d r þ12d W d r
2;ð8a ÞL.S.Ma,T.J.Wang /International Journal of Solids and Structures 40(2003)3311–33303313
e 0h ¼U
r ð8b Þ
with U being the displacement in r direction.Variations of the curvature j r and j h in the mid-plane of the plate (z ¼0)can be calculated as,
j r ¼Àd 2W d r
2;ð9a Þj h ¼À1r d W d r :ð9b ÞThe temperature difference T ðz Þfrom the stress free state is governed by the following well-known heat transfer equation Àd d z K ðz Þd T ðz Þd z
¼0ð10Þwith the boundary conditions T ðh =2Þ¼T 1and T ðÀh =2Þ¼T 2.The thermal conductivity coefficient K ðz Þin Eq.(10)follows the distribution law of Eqs.(1)and (2),namely,
K ðz Þ¼ðK m ÀK c ÞV m þK c :
It is easily to obtain from Eq.(10)that
T ðz Þ¼T 2þðT 1ÀT 2ÞZ z Àh =2d z K ðz Þ Z h =2Àh =2
d z K ðz Þ:ð11ÞFrom Eqs.(6)and (7),on
e obtains N r N h ¼A 11A 12A 12A 22 e 0r e 0h
þB 11B 12B 12B 22 j r j h ÀN T r N T h ;ð12ÞM r
M h ¼B 11B 12B 12B 22 e 0r e 0h þD 11D 12D 12D 22 j r j h ÀM T r M T h ;ð13Þ
where A ij ,B ij and D ij are stiffness coefficients of the plate and can be calculated as
ðA ij ;B ij ;D ij Þ¼
Z h =2
Àh =2Q ij ð1;z ;z 2Þd z ð14Þ
with
Q 11¼Q 22¼
E ðz Þ1Àm 2
;Q 12¼m Q 11:The forces and moments in Eqs.(12)and (13)induced by thermal load can be calculated as
N T r ¼
Z h =2Àh =2E ðz Þ1Àm a ðz ÞT ðz Þd z ;ð15a ÞN T h ¼Z
h =2Àh =2
E ðz Þ1Àm a ðz ÞT ðz Þd z ;ð15b ÞM T r ¼Z h =2Àh =2
E ðz Þ1Àm a ðz ÞT ðz Þz d z ;ð15c Þ
3314L.S.Ma,T.J.Wang /International Journal of Solids and Structures 40(2003)3311–3330
M T
h ¼
Z h=2
Àh=2
EðzÞ
1Àm
aðzÞTðzÞz d z:ð15dÞ
From Eqs.(3)–(5),(8),(9),(12),(13)and(15),one then obtains the governing equations expresd in terms of the displacements
A11
d2U
d r2
"
þ
1
r
d U
d r
À
U
r2
þ
d2W
d r2
d W
d r
þ
1Àm
2r
d W
d r
2#
¼B11
d3W
d r3
þ
1
r
d2W
d r2
À
1
r2
d W
d r
;ð16Þ
D11
d4W
d r4
þ
2
r
d3W
d r3
À
1
r2
d2W
d r2
þ
1
r3
d W
d r
¼A11
d U
d r
"
þ
m
r
Uþ
1
2
d W
d r
2#d2W
d r2
þA11m
d U
d r
"
þ
1
r
Uþ
m
2
d W
d r
2#1
r
d W
d r
þB11
d3U
d r3
þ
2
r
d2U
d r2
À
1
r2
d U
d r
þ
1
r3
U
þB11
d3W
d r3
þ
2À3m
r
d2W
d r2
À
1
r2
d W
d r
d W
d r
ÀN T
r
1
r
d W
d r
þ
d2W
d r2
þq:ð17Þ
The continuity conditions at the center of plate results in W beingfinite,and
U¼d W
¼0;lim
r!0
d3W
3
þ
1d2W
2
¼0at r¼0:
In what follows,two types of boundary conditions are examined.
Ca1.The plate edge is clamped and immovable in r direction.Such that the boundary conditions can be expresd as
U¼W¼d W
d r
¼0at r¼b:ð18Þ
Ca2.The plate edge is simply supported and immovable in r direction.Such that the boundary conditions can be expresd as
U¼W¼0;B11
d U
d r
"
þ
m
r
Uþ
1
2
d W
d r
2#
ÀD11
d2W
d r2
þ
m
r
d W
d r
ÀM T
r
¼0at r¼b:ð19Þ
Now,a nonlinear bending problem is formulated.If q¼0,the problem reduces to a nonlinear buckling problem.It is difficult to solve such a nonlinear problem due to the inhomogeneity of material.For con-venience,the following dimensionless parameters are introduced,
x¼r
b
;w¼
W
h
;u¼
Ub
h2
;F1¼
B11
D11
h;F2¼
A11
D11
h2;F3¼
B11
hA11
;N¼
N T
r
b2
D11
;M¼
M T
r
b2
D11h
;
k¼12b2
h2
ð1þmÞa c T2;Q¼
qb4
D11h
;E r¼
E m
E c
:
Such that the dimensionless governing equations and boundary conditions of the plate can be expresd as
d2u d x2þ
1
x
d u
d x
À
u
x2
þ
d2w
d x2
d w
d x
þ
1Àm
2x
d w
d x
2
¼F3
d3w
d x3
þ
1
x
d2w
d x2
À
1
x2
d w
d x
;ð20ÞL.S.Ma,T.J.Wang/International Journal of Solids and Structures40(2003)3311–33303315
