Coupled thermoelasticity of functionally
graded cylindrical shells
A.Bahtui
a,*
,M.R.Eslami
b
a Postgraduate Campus,Tehran South Branch,Islamic Azad University,Tehran,Iran b
Mechanical Engineering Department,Amirkabir University of Technology,Tehran,Iran
Available online 13October 2005
Abstract
The coupled thermoelastic respon of a functionally graded circular cylindrical shell is studied.The co
upled thermo-elastic and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell sub-jected to thermal shock load.A cond-order shear deformation shell theory that accounts for the transver shear strains and rotations is considered.Including the thermo-mechanical coupling and rotary inertia,a Galerkin finite element formulation in space domain and the Laplace transform in time domain are ud to formulate the problem.The inver Laplace transform is obtained using a numerical algorithm.The shell is graded through the thickness assuming a volume fraction of metal and ceramic,using a power law distribution.The results are validated with the known data in the literature.
Ó2005Published by Elvier Ltd.
Keywords:Coupled thermoelasticity;Cylindrical shell;Functionally graded material;Thermal shock
1.Introduction
When the characteristic times of structural and thermal disturbances are of comparable magnitudes,the equations of motion of a structure are coupled with the energy equation and the solution of the coupled sys-tem of equations provides the stress and temperature fields in the shell.Eslami et al.(1994)prented the cou-pled thermoelasticity solution of a long circular cylindrical
thin shell.They ud a linear temperature distribution across the thickness of the shell,employing the classical coupled theory of thermoelasticity.Rao and Sinha (1997)applied a coupled thermo-structural analysis of laminated composite beams to compare the results for steady,uncoupled and coupled theories to show the effects of laminate stacking quences and structural boundary conditions.Reddy and Chin (1998)prented a thermo-mechanical analysis of function-ally graded thick cylinders and thin plates.They assumed temperature-dependent properties in order to obtain more realistic results.Eslami et al.(1999)studied the coupled thermoelasticity of shells of revolution.
0093-6413/$-e front matter Ó2005Published by Elvier Ltd.doi:10.hrescom.2005.09.003
*
Corresponding author.
E-mail address:ali. (A.
Bahtui).
Mechanics Rearch Communications 34(2007)
1–18
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MECHANICS
RESEARCH COMMUNICATIONS
They employed the Flu ¨gge cond-order shell theory with the linear temperature distribution across the shell thickness,and studied the effect of normal stress and coupling.Sharma and Sharma (1999)prented the ther-moelastic respon of thick axisymmetric solid plate subjected to sudden lateral loading and thermal shock.Shiari et al.(2003)extended the Eslami Õs work to analyze the thermo-mechanical behavior of the multilayer orthotropic composite cylindrical shells subjected to the thermal shock loads.
He et al.(2001)studied a finite element formulation bad on the classical laminated plate theory for the shape and vibration control of the functionally graded material (FGM)plates with integrated piezoelectric nsors and actuators.Ng et al.(2002)prented a flat-shell element for the active control of FG shells through integrated piezoelectric nsor/actuator layers.Liew et al.(2002b)studied
an efficient meshfree formulation bad on the first-order shear deformation theory for the static analysis of laminated composite beams and plates with integrated piezoelectric layers.Liew et al.(2001)prented an efficient finite element formulation bad on a FSDT for the active control of FG plates with integrated piezoelectric nsor/actuator layers sub-jected to a thermal gradient.Liew et al.(2002a)derived a generic static and dynamic finite element formulation for the modeling and control of piezoelectric shell laminates under coupled displacement,temperature and
Nomenclature
c specific heat conduction E Young Õs modulus f c ceramic-pha volume fraction f m metal-pha volume fraction F ef effective material property of FGM h in ,h out inside,outside convective coefficients h shell thickness k volume fraction index K thermal conduction coefficient L length NE number of elements p z (t )lateral force
q in ,q out inside,outside heat flux R mean radius s Laplace variable t time variable T shell temperature T 0mean shell temperature T 1gradient through the thickness T a reference temperature T 1ambient temperature
u 0,v 0,w 0displacement components V c ceramic-pha volume V m metal-pha volume x ,h ,z sh
ell coordinates a thermal expansion coefficient b ij coupling coefficient tensor D T temperature change n shape function variable m Poisson Õs ratio /ðe Þ1;/ðe Þ
3shape functions q mass density
w x ,w h middle plane tangents rotations
2 A.Bahtui,M.R.Eslami /Mechanics Rearch Communications 34(2007)1–18
electric potentialfields.He et al.(2002)developed a genericfinite element formulation for the static and dynamic control of FG shells with piezoelectric nsor and actuator layers.Yang et al.(2004a)investigated the non-linear bending behavior of FG plates that are bonded with piezoelectric actuator layers and subjected to transver loads and a temperature gradient bad on ReddyÕs higher-order shear deformation plate theory. Liew et al.(2003b)studied the postbuckling behavior of piezoelectric FG rectangular plates that are subjected to the combined action of uniform temperature change,in-plane forces,and constant applied actuator voltage. Yang et al.(2003)prented a large amplitude vibration analysis of pre-stresd FG laminated plates that are compod of a shear deformable functionally graded layer and two surface-mounted piezoelectric actuator layers.
Yang et al.(2004b)solved a dynamic stability analysis of symmetrically laminated FG rectangular plates with general out-of-plane supporting conditions,subjected to a uniaxial periodic in-plane load and undergoing uniform temperature change.Kitipornchai et al.(2004)prented the nonlinear vibration of imperfect shear deformable laminated rectangular plates comprising a homogeneous substrate and two layers of functionally graded materials.Bagri and Eslami(2004)analyzed generalized coupled thermoelasticity of disks bad on the Lord–Shulman model.A number of other related topics may be found in references(Shen et al.,2004;Chen and Liew,2004;Eslami and Vahedi,1992;Hata,1992;Li et al.,1986;Liew et al.,2003a,2004;Zhang et al., 1993).
The prent work deals with the coupled thermoelasticity problem of a Titanium–Zirconia functionally graded cylindrical shell under impulsive thermal shock load.The material properties are graded along the thickness direction according to a volume fraction power law distribution.The governing equations are bad on the cond-order shear deformation shell theory,including the rotary inertia term.The classical linear ther-moelastic theory is considered,and the Galerkinfinite element formulation is ud.The governing equations are transformed into the Laplace domain,where the Galerkinfinite element method is ud to obtain the solu-tion in the space domain.The inversion of Laplace transform into the physical time domain is obtained numerically.Thefinite element analysis is
carried out with two element types C0-continuous and C1-contin-uous,where their results are compared.The numerical results for the temperature,displacement,and the ther-mal stress are shown infigures.The results are reduced to the shells with isotropic material and validated with the known data in the literature.
2.Analysis
The material properties of the functionally graded shell,such as YoungÕs modules E(z),thermal expansion coefficient a(z),thermal conduction coefficient K(z),specific heat conduction c(z),and density q(z)must be described across the shell thickness,where z is the thickness coordinate(Àh/26z6h/2),and h is the thickness of shell.We assume that the functionally graded shell is comprid of metal-pha and ceramic-pha.Let us assume that V m and V c reprent the volumes of metal and ceramic phas,respectively.Let the volume frac-tion of each constituent material be denoted by
f m¼
V m
V mþV c
;f
c
¼
V c
V mþV c
.
Here,f m and f c are the volume fractions of metal and ceramic of FGM,respectively,and satisfy the following equation:
f mþf c¼1.
The volume fraction is a spatial function.Using the combination of the functions,the effective material properties of functionally graded materials may be expresd as
F efðzÞ¼F m f mþF c f c;
where F ef is the effective material property of functionally graded material,and F m and F c are the persistent material properties of each pha.The volume fraction is assumed to follow a power law function as(Reddy and Chin,1998)
A.Bahtui,M.R.Eslami/Mechanics Rearch Communications34(2007)1–183
f m¼
2zþh
2h
k
;
f c¼1Àf m;
ð1Þ
where volume fraction index k reprents the material variation profile through the shell thickness,wh
ich is always greater than or equal to zero,and may be varied to obtain the optimum distribution of the constituent materials.The value of k equal to zero reprents a fully metal and infinity reprents a fully ceramic shell.
From Eq.(1),the effective material properties of a functionally graded cylindrical shell may be written as (Reddy and Chin,1998)
Table1
Material properties of Titanium and Zirconia
Metal:Ti–6Al–4V Ceramic:ZrO2
E=66.2(GPa)E=117.0(GPa)
m=0.321m=0.333
a=10.3·10À6(1/K)a=7.11·10À6(1/K)
q=4.41·103(kg/m3)q=5.6·103(kg/m3)
K=18.1(W/m K)K=2.036(W/m K)
c=808.3(J/kg K)c=615.6(J/kg K)
4 A.Bahtui,M.R.Eslami/Mechanics Rearch Communications34(2007)1–18
A.Bahtui,M.R.Eslami/Mechanics Rearch Communications34(2007)1–185
EðzÞ¼E cþE mc f m;
aðzÞ¼a cþa mc f m;
KðzÞ¼K cþK mc f m;
ð2ÞqðzÞ¼q
þq mc f m;
c
cðzÞ¼c cþc mc f m;
where
E mc¼E mÀE c;
a mc¼a mÀa c;
K mc¼K mÀK c;
ð3Þ¼q mÀq c;
q
mc
c mc¼c mÀc c.
It is assumed that the PoissonÕs ratio m is constant across the shell thickness.
2.1.Strain–displacement relations
Consider a functionally graded circular cylindrical shell offinite length L,wall thickness h,and mean radius
R.The cylindrical coordinates(x,h,z)are considered along the axial,circumferential,and radial directions,
