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Transient Thermoelastic Analysis for a MultilayeredHollow Circular Disk with Piecewise Power LawNonhomogeneityYoshihiro Ootao a & Yoshinobu Tanigawa aa Department of Mechanical Engineering, Graduate School of Engineering, Osaka PrefectureUniversity, Sakai, JapanVersion of record first published: 08 Feb 2012.
To cite this article: Yoshihiro Ootao & Yoshinobu Tanigawa (2012): Transient Thermoelastic Analysis for a Multilayered HollowCircular Disk with Piecewise Power Law Nonhomogeneity, Journal of Thermal Stresses, 35:1-3, 75-90
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Journal of Thermal Stresses, 35: 7590, 2012Copyright Taylor & Francis Group, LLCISSN: 0149-5739 print/1521-074X onlineDOI: 10.1080/01495739.2012.637749
TRANSIENT THERMOELASTIC ANALYSIS FOR AMULTILAYERED HOLLOW CIRCULAR DISK WITHPIECEWISE POWER LAW NONHOMOGENEITY
Yoshihiro Ootao and Yoshinobu TanigawaDepartment of Mechanical Engineering, Graduate School of Engineering,Osaka Prefecture University, Sakai, Japan
This article is concerned with the theoretical treatment of transient thermoelasticproblem involving a multilayered hollow circular disk with piecewise power lawnonhomogeneity due to uniform heat supply from inner and outer surfaces. Themultilayered hollow circular disk is cooled from the upper and lower surfaces ofthe each layer with constant heat transfer coefcient. The thermal conductivity, theYoungs modulus and the coefcient of linear thermal expansion of each layer areexpressed as power functions of the radial coordinate, and their values continue on theinterfaces. We obtain the exact solution for the one-dimensional temperature change ina transient state, and in-plane thermoelastic response under the state of plane stress.Some numerical results for the temperature change, the displacement and the stressdistributions are shown in gures.
Keywords: Functionally graded material; Multilayered circular disk; Power law nonhomogeneity;Thermoelasticity; Transient state
INTRODUCTION
As a new nonhomogeneous material system, Functionally graded materials(FGMs) were proposed in Japan at rst. FGMs are those in which two or moredifferent material ingredients change continuously and gradually along the certaindirection. In recent years, the concept of FGMs has been applied in many industrialelds [1, 2]. When FGMs are used under high temperature conditions or aresubjected to several thermal loading, it is necessary to analyze the thermal stressproblems for FGMs.
It is well-known that thermal stress distributions in a transient state canshow large values compared with the one in a steady state. Therefore, the analysisof transient thermoelastic problem for FGMs becomes important. Because thegoverning equations for the temperature eld and the associate thermoelastic eldof FGMs become of nonlinear form in generally, the analytical treatment is very
Received 30 April 2011; accepted 17 June 2011.This article appears in a special triple issue of the Journal of Thermal Stresses dedicated to
Professor Naotake Noda on the occasion of his 65th birthday and his retirement.Address correspondence to Yoshihiro Ootao, Department of Mechanical Engineering, Graduate
School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai 599-8531, Japan.E-mail: [email protected]
75
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76 Y. OOTAO AND Y. TANIGAWA
difcult. As the analytical treatment of the thermoelastic problems of FGMs, thereare two pieces of treatment mainly. One is introducing the theory of laminatedcomposites, which have a number of homogeneous layers along the thicknessdirection. Examples using the theory of laminated composites are as follows.
We analyzed theoretically the transient thermal stress problems of severalanalytical models, i.e., hollow cylinders [35], plates [6, 7] and hollow spheres [8, 9].Sugano et al. reported an approximate three-dimensional analysis of thermalstresses in a nonhomogeneous plate with temperature change and nonhomogeneousproperties only in the thickness direction [10] and a one-dimensional analysis oftransient thermal stress in a circular plate with arbitrary variation of heat-transfercoefcient [11].
The other analytical treatment is the exact analysis under the assumptionthat the material properties are given in the specic functions containing thevariable of the thickness coordinate without using the laminated composite model.Examples of the rectangular coordinates are as follows. Sugano analyzed exactlyone-dimensional transient thermal stresses of nonhomogeneous plate where thethermal conductivity and Youngs modulus vary exponentially, whereas Poissonsratio and the coefcient of linear thermal expansion vary arbitrarily in the thicknessdirection [12].
Vel and Batra analyzed the three-dimensional transient thermal stresses ofthe functionally graded rectangular plate [13]. We analyzed the transient thermalstress problems of a functionally graded thick strip [14] and a functionally gradedrectangular plate [15], where the thermal conductivity, the coefcient of linearthermal expansion and Youngs modulus vary exponentially in the thicknessdirection, due to nonuniform heat supply.
Examples of the cylindrical coordinates are as follows. We obtained the one-dimensional solution for transient thermal stresses of a functionally graded hollowcylinder whose material properties vary with the power product form of radialcoordinate variable [16]. Zhao et al. analyzed the one-dimensional transient thermo-mechanical behavior of a functionally graded solid cylinder, whose thermoelasticconstants vary exponentially through the thickness [17]. Shao et al. analyzed one-dimensional transient thermo-mechanical behavior of functionally graded hollowcylinders, whose thermoelastic constants are expressed as Taylors series [18]. Forthe case of nonuniform distributed heating, Shao et al. obtained the analyticalsolutions for transient thermomechanical response of functionally graded cylindricalpanels [19] and functionally graded hollow cylinders [20]. We obtained the two-dimensional analytical solution for the transient thermal stresses of a functionallygraded cylindrical panel whose material properties vary with the power productform of radial coordinate variable [21]. As the examples of the circular plates, Pengand Li analyzed the steady thermal stress problem in rotating functionally gradedhollow circular disks [22].
Go et al. analyzed the steady thermal stress problem in a rotatingfunctionally graded hollow circular disk by nite element method [23]. However,these studies discuss the thermoelastic problems of one-layered FGM models,which have the big limitation of nonhomogeneity. On the other hand, thearbitrary nonhomogeneity can be expressed in the theory of laminated compositesapproximately, but the material properties are discontinuous on the interfaces. Thetransient thermoelastic problem for a multilayered hollow cylinder with piecewise
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THERMOELASTICITY FOR A MULTILAYERED HOLLOW CIRCULAR DISK 77
power law nonhomogeneity as a new FGM model with arbitrary properties wasanalyzed [24]. To the authors knowledge, the exact analysis for a transientthermoelastic problem of multilayered hollow circular disk has not been reported.
From the viewpoint of above-mentioned, we analyze the transientthermoelastic analysis for a multilayered hollow circular disk with piecewise powerlaw nonhomogeneity as a new FGM model with arbitrary properties.
ANALYSIS
Consider a multilayered hollow circular disk with piecewise power lownonhomogeneity. The thermal conductivity, the Youngs modulus and thecoefcient of linear thermal expansion of each layer are expressed as powerfunctions of the radial coordinate, and their values continue on the interfaces. Thehollow circular disks inner and outer radii are designated ra and rb, respectively.Moreover, ri is the outer radius of ith layer. The thickness of the hollow circulardisk is represented by B.
Heat Conduction Problem
The multilayered hollow circular disk is assumed to be initially at zerotemperature and is suddenly heated from the inner and outer surfaces bysurrounding media of constant temperatures Ta and Tb with relative heat transfercoefcients ha and hb. The multilayered hollow circular disk is cooled from theupper and lower surfaces of the ith layer by surrounding media of zero temperaturewith heat transfer coefcient is. Then the temperature distribution shows a one-dimensional distribution, and the transient heat conduction equation for the ithlayer is taken in the following forms:
ciiTit
= 1r r
[irr
Tir
] 2si
BTi (1)
The thermal conductivity i and the heat capacity per unit volume cii in each layeris assumed to take the following forms:
ir = 0i(
r
ri1
)mi cii = const (2)
where
mi =ln0i+1/
0i
lnri/ri1(3)
Substituting the Eqs. (2) and (3) into the Eq. (1), the transient heat conductionequation in dimensionless form is
Ti
= 0i
ciirmii1
[mi + 1rmi1
Tir
+ rmi 2Tir2
] 2Hsi
ciiBTi i = 1 2 N (4)
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78 Y. OOTAO AND Y. TANIGAWA
The initial and thermal boundary conditions in dimensionless form are
= 0 Ti = 0 i = 1 2 N (5)
r = raT1r
HaT1 = HaTa (6)r = ri Ti = Ti+1 i = 1 2 N 1 (7)
r = riTir
= Ti+1r
i = 1 2 N 1 (8)
r = 1 TNr
+HbTN = HbTb (9)
In Eqs. (3)(9), we have introduced the following dimensionless values:
TiTaTb =Ti Ta Tb
T0 r ri raB =
r ri ra B
rb = 0t
r2b 0 =
0c00
cii =ciic00
i 0i =
i 0i
0 HaHb = ha hbrb Hsi = sirb/0 (10)
where Ti is the temperature change; t is time; and T0, 0, and c00 are typicalvalues of temperature, thermal conductivity and heat capacity per unit volume,respectively. Introducing the Laplace transformation with respect to the variable ,the solution of Eq. (4) can be obtained so as to satisfy the conditions (5)(9). Thissolution is shown as follows:
Ti =1Frmi/2
[Ai Ii ir1mi/2+Bi Ki ir1mi/2]
+j=1
201B1j2m12rmi/221j
012m12B + 8Hs1rm1a 1j
exp
([012m122ij
4c11rm1a
+ 2Hs1c1iB
]
)
[AiJi
(K1i
2ij + K2ir1mi/2
)+BiYi
(K1i
2ij + K2ir1mi/2
)]if mi = 0 2
(11)
where Ix and Kx are the modied Bessel functions of the rst and second kindof order x, respectively. Jx and Yx are the Bessel functions of the rst andsecond kind of order x, respectively. And, and F are the determinants of 2N 2N matrix [akl] and [ekl], respectively; the coefcients Ai and Bi are dened as thedeterminant of the matrix similar to the coefcient matrix [akl], in which the 2i1th column or 2ith column is replaced by the constant vector ck, respectively.
Similarly, the coefcients Ai and Bi are dened as the determinant of thematrix similar to the coefcient matrix [ekl], in which the 2i 1th column or 2ithcolumn is replaced by the constant vector ck, respectively. The nonzero elementsof the coefcient matrices [akl], [ekl] and the constant vector ck are given as
a11 = rm12
a
{[(1 m1
2
)(1
m12m1
)r1a Ha
]J1
(
1r
1 m12a
)
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THERMOELASTICITY FOR A MULTILAYERED HOLLOW CIRCULAR DISK 79
1(1 m1
2
)r m12a J1+1
(
1r
m12a
)}
a12 = rm12
a
{[(1 m1
2
)(1
m12m1
)r1a Ha
]Y1
(
1r
1 m12a
)
1(1 m1
2
)r m12a Y1+1
(
1r
m12a
)}
a2N2N1 =[(
1 mN2
)(N
mN2mN
)+Hb
]JN N N
(1 mN
2
)JN+1N
a2N2N =[(
1 mN2
)(N
mN2mN
)+Hb
]YN N N
(1 mN
2
)YN+1N
(12)
a2i2i1 = rmi2
i Ji
(
ie
1 mi2) a2i2i = r
mi2
i Yi
(
ie
1 mi2)
a2i2i+1 = rmi+12
i Ji+1
(
i+1r
1 mi+12i
) a2i2i+2 = r
mi+12
i Yi+1
(
i+1r
1 mi+12i
)
a2i+12i1 =(1 mi
2
)r mi2i
[(i
mi2mi
)r1i Ji
(
ir
1 mi2i
)ir
mi2
i Ji+1
(
ir
1 mi2i
)]
a2i+12i =(1 mi
2
)r mi2i
[(i
mi2mi
)r1i Yi
(
ir
1 mi2i
)ir
mi2
i Yi+1
(
ir
1 mi2i
)]
a2i+12i+1 = (1 mi+1
2
)r mi+12i
[(i+1
mi+12mi+1
)r1i Ji+1
(
i+1r
1 mi+12i
)
i+1rmi+12
i Ji+1+1
(
i+1r
1 mi+12i
)]
a2i+12i+2 = (1 mi+1
2
)r mi+12i
[(i+1
mi+12mi+1
)r 1i Yi+1
(
i+1r
1 mi+12i
)
i+1rmi+12
i Yi+1+1
(
i+1r
1 mi+12i
)] i = 1 2 N 1 (13)
e11 = rm12
a
{[(1 m1
2
)(1
m12m1
)r1a Ha
]I1
(1r
1 m12a
)
+ 1(1 m1
2
)r m12a I1+1
(1r
m12a
)}
e12 = rm12
a
{[(1 m1
2
)(1
m12m1
)r1a Ha
]K1
(1r
1 m12a
)
1(1 m1
2
)r m12a K1+1
(1r
m112a
)}
e2N2N1 =[(
1 mN2
)(N
mN2mN
)+Hb
]IN N+ N
(1 mN
2
)IN+1N
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80 Y. OOTAO AND Y. TANIGAWA
e2N2N =[(
1 mN2
)(N
mN2mN
)+Hb
]KN N N
(1 mN
2
)KN+1N
(14)
e2i2i1 = rmi2
i Ii(ie
1 mi2) e2i2i = r
mi2
i Ki(ie
1 mi2)
e2i2i+1 = rmi+12
i Ii+1
(i+1r
1 mi+12i
) e2i2i+2 = r
mi+12
i Ki+1
(i+1r
1 mi+12i
)
e2i+12i1 =(1 mi
2
)r mi2i
[(i
mi2mi
)r1i Ii
(ir
1 mi2i
)+ir
mi2
i Ii+1
(ir
1 mi2i
)]
e2i+12i =(1 mi
2
)r mi2i
[(i
mi2mi
)r1i Ki
(ir
1 mi2i
)ir
mi2
i Ki+1(ir
1 mi2i
)]
e2i+12i+1 = (1 mi+1
2
)r mi+12i
[(i+1
mi+12mi+1
)r 1i Ii+1
(i+1r
1 mi+12i
)
+ i+1rmi+12
i Ii+1+1
(i+1r
1 mi+12i
)]
e2i+12i+2 = (1 mi+1
2
)r mi+12i
[(i+1
mi+12mi+1
)r1i Ki+1
(i+1r
1 mi+12i
)
i+1rmi+12
i Ki+1+1
(i+1r
1 mi+12i
)] i = 1 2 N 1 (15)
c1 = HaTa c2N = HbTb (16)
In Eq. (11), K1i, K2i, 1j, i, and i are
K1i =012m12ciirmii10i 2mi2c11rm1a
K2i =8ciir
mii1
0i 2mi2B(Hs1c11
Hsicii
)
1j =d
d1
1=1j
i =
8Hsirmii1
0iB2mi2 i =
mi2mi (17)
and 1j represent the jth positive roots of the following transcendental equation
1 = 0 (18)
In Eqs. (12) and (13), the relations between i and 1 are
i =K1i
21 + K2i i = 1 2 N (19)
Thermoelastic Problem
Let analyze the transient thermoelasticity of a multilayered circular diskas a plane stress problem. The displacement-strain relations are expressed in
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THERMOELASTICITY FOR A MULTILAYERED HOLLOW CIRCULAR DISK 81
dimensionless form as follows:
rri = urir i =urir
(20)
where a comma denotes partial differentiation with respect to the variable thatfollows. The constitutive relations are expressed in dimensionless form as follows:{
rrii
}= Ei
1+ i1 i[1 ii 1
]{rri i
} iEiTi
1 2i(21)
The equilibrium equation is expressed in dimensionless form as follows:
rrir +1rrri i = 0 (22)
The Youngs modulus Ei, the coefcient of linear thermal expansion i andPoissons ratio i are assumed to take the following forms:
Eir = E0i(
r
ri1
)li ir = 0i
(r
ri1
)bi i = const (23)
wherev
li =lnE0i+1/E0i lnri/ri1
bi =ln0i+1/
0i
lnri/ri1(24)
In Eqs. (20)(24), the following dimensionless values are introduced:
kli =kli
0E0T0 kli =
kli0T0
i 0i =
i 0i
0
EiE0i =Ei E
0i
E0 uri =
uri0T0rb
(25)
where kli are the stress components, kli are the strain components, uri is thedisplacement in the radial direction, and 0 and E0 are the typical values ofthe coefcient of linear thermal expansion and Youngs modulus, respectively.Substituting Eqs. (20), (21) and (23) into Eq. (22), the displacement equation ofequilibrium is written as
urir r +li + 1r
urir + ili 1urir2 =1+ i0i
rbii1
li + birbi1Ti + rbiTir (26)
If the inner and outer surfaces are traction free, and the interfaces of the each layerare perfectly bonded, then the boundary conditions of inner and outer surfaces andthe conditions of continuity on the interfaces can be represented as follows:
r = ra rr1 = 0 (27)r = ri rri = rri+1 uri = uri+1 (28)
r = 1 rrN = 0 (29)
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82 Y. OOTAO AND Y. TANIGAWA
We assume the solution of Eq. (26) in the following form.
uri = uric + urip (30)
In Eq. (30), the rst term on the right side gives the homogeneous solution andthe second term of right side gives the particular solution. We now consider thehomogeneous solution, and introduce the following equation:
r = exps (31)
Changing a variable with the use of Eq. (31), the homogeneous expression ofEq. (26) is
D2 + liD 1 iliuric = 0 (32)
where
D = dds
(33)
We now introduce the following expression:
Hi = l2i + 41 ili (34)
Because Hi is positive in generally, there are two distinct real roots as follows:
i1 = li +Hi/2 i2 = li
Hi/2 (35)
The homogeneous solution uricr is given by the following expression:
uric = A1iri1 + A2iri2 (36)
In Eq. (36), Ai1 and Ai2 are unknown constants.To obtain the particular solution, we use the series expansions of the Bessel
functions and the modied Bessel functions. Because the order i of the Besselfunctions in Eq. (11) is not integer in general, Eq. (11) can be written as thefollowing expression.
Tir =n=0
anirw1i + bnirw2i (37)
where
w1i =122mi2n+ imi
w2i =122mi2n imi (38)
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THERMOELASTICITY FOR A MULTILAYERED HOLLOW CIRCULAR DISK 83
ain =1F
(Ai Bi
2 sin i
)1
n!i + n+ 1(i2
)2n+i
+j=1
201B1j2m1221j
012m12B + 8Hs1rm1a 1j
exp([012m122ij
4c11rm1a
+ 2Hs1c1iB
]
)
(Ai +
cos isin i
Bi)
1nn!i + n+ 1
K1i
21j + K2i2
2n+i
bin =1FBi
2 sin i 1n!i + n+ 1
(i2
)2ni
+j=1
201B1j2m1221j
012m12B + 8Hs1rm1a 1j
exp([012m122ij
4c11rm1a
+ 2Hs1c1iB
]
)
Bi1
sin i 1
n
n!i + n+ 1K1i
21j + K2i2
2ni
(39)
The particular solution urip is
uripri =n=0
fanirw1i+bi+1 + fbnirw2i+bi+1 (40)
Expressions for fani and fbni in Eq. (40) are omitted here for the sake ofbrevity. Then, the stress components can be evaluated by substituting Eq. (30) intoEq. (20), and later into Eq. (21). The unknown constants in Eq. (36) are determinedso as to satisfy the boundary conditions (27)(29).
NUMERICAL RESULTS
We consider the functionally graded materials composed of titanium alloy(Ti-6Al-4V) and zirconium oxide (ZrO2). We assume that the hollow circular disk isheated from the outer surface (zirconium oxide 100%) by surrounding media. Thematerial of the inner surface is titanium alloy 100%. The material properties gi ofthe interface between ith layer and (i+ 1)th layer are assumed as follows:
gi = ga + gb gaci 0 ci 1 i = 1 2 N 1 (41)
where ga is the material property of the inner surface, and gb is the material propertyof the outer surface. The numerical parameters of heat conduction, shape and ci are
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84 Y. OOTAO AND Y. TANIGAWA
presented as follows:
Ha = 10 Hb = 100 Ta = 0 Tb = 10 B = 005 (42)Case 1: Hsi = 01 N = 2 ra = 02 r1 = 06 c1 = 05 (43)Case 2: Hsi = 01 N = 2 ra = 02 r1 = 06 c1 = 01 05 09 (44)Case 3: Hsi = 01 N = 3 ra = 02 r1 = 14/3 r2 = 22/3
c1 = 01 c2 = 02 05 09 (45)Case 4: Hsi = 01 N = 1 ra = 02 (46)Case 5: Hsi = 10 N = 2 ra = 02 r1 = 06 c1 = 05 (47)
The material constants for titanium alloy (Ti-6Al-4V) are taken as,
= 261 106 m2/s c = 5377 J/(kg K) = 4420 kg/m3 = 62W/(m K) = 89 106 1/K E = 1058GPa = 03 (48)
for zirconium oxide (ZrO2),
= 106 106 m2/s c = 4614 J/kg K = 3657 kg/m3 = 178 10W/(m K) = 87 106 1/K E = 1164 10GPa = 03 (49)
The typical values of material properties such as 0, 0, 0 and E0 used to normalizethe numerical data, are based on those of zirconium oxide.
Figure 1 Variation of temperature change in the radial direction (Case 1).
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THERMOELASTICITY FOR A MULTILAYERED HOLLOW CIRCULAR DISK 85
Figure 2 Variation of displacement ur in the radial direction (Case 1).
The numerical results for Case 1 are shown in Figures 13. Figure 1 showsthe variation of temperature change along the radial direction. Figure 2 shows thevariation of displacement ur along the radial direction. From Figures 1 and 2, thetemperature and displacement rise as time proceeds and are greatest in the steadystate. Figures 4a and 4b show the variations of thermal stresses rr and alongthe radial direction, respectively. From Figure 3a, the maximum tensile stress occursin the transient state inside the hollow circular disk. From Figure 3b, the largecompressive stress occurs on the heated surface and the tensile stress occurs nearthe inner surface.
Figure 3 Variation of thermal stresses in the radial direction (Case 1): (a) normal stress rr and(b) normal stress .
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86 Y. OOTAO AND Y. TANIGAWA
Figure 4 Variation of thermal stresses in the radial direction (Case 2, = : (a) normal stress rrand (b) normal stress .
To assess the inuence of the material property distribution for two-layeredFGM model, the numerical results for Case 2 are shown in Figure 4. Figures 4aand 4b show the variations of thermal stresses rr and in the steady state,respectively. It can be seen from Figure 4 that the maximum values of the thermalstresses rr and decrease when the parameter c1 decreases.
To assess the inuence of the material property distribution for three-layeredFGM model, the numerical results for Case 3 are shown in Figure 5. Figures 5aand 5b show the variations of thermal stresses rr and in the steady state,
Figure 5 Variation of thermal stresses in the radial direction (Case 3, = : (a) normal stress rrand (b) normal stress .
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THERMOELASTICITY FOR A MULTILAYERED HOLLOW CIRCULAR DISK 87
Figure 6 Variation of thermal stresses in the radial direction (Case 4): (a) normal stress rr and (b)normal stress .
respectively. It can be seen from Figure 5 that the maximum values of the thermalstresses rr and decrease when the parameter c2 decreases.
To assess the inuence of the functional grading, the numerical results forCase 4, i.e., one-layered FGM model, are shown in Figure 6. Figures 6a and 6b showthe variations of thermal stresses rr and along the radial direction, respectively.In comparison with the numerical results for Case 2 and Case 3, it is possible todecrease the maximum values of thermal stresses rr and using the multilayeredFGM model with piecewise power law nonhomogeneity.
Figure 7 Variation of temperature change in the radial direction (Case 5).
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88 Y. OOTAO AND Y. TANIGAWA
Figure 8 Variation of displacement ur in the radial direction (Case 5).
To assess the inuence of the relative heat transfer coefcient of upper andlower sides, the numerical results for two-layered FGM model, are shown inFigures 79. Figure 7 shows the variation of temperature change along the radialdirection. Figure 8 shows the variation of displacement ur along the radial direction.Figures 9a and 9b show the variations of thermal stresses rr and along theradial direction, respectively. In comparison with the numerical results for Case 2,it can be seen that the values of the temperature, displacement and thermal stress
Figure 9 Variation of thermal stresses in the radial direction (Case 5): (a) normal stress rr and(b) normal stress .
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THERMOELASTICITY FOR A MULTILAYERED HOLLOW CIRCULAR DISK 89
rr decrease when the Biots number Hsi increases. From Figures 3b and 9b, themaximum tensile stress of decreases when the Biots number Hsi increases, whilethe maximum compressive stress of doesnt decrease.
CONCLUSION
In the present article, we analyzed the transient thermoelastic probleminvolving a multilayered hollow circular disk with piecewise power lawnonhomogeneity due to uniform heat supply from inner and outer surfaces. Themultilayered hollow circular disk is cooled from the upper and lower surfaces ofthe each layer with constant heat transfer coefcient. The thermal conductivity,the Youngs modulus and the coefcient of linear thermal expansion of each layerare expressed as power functions of the radial coordinate in the radial direction,and their values continue on the interfaces. We obtained the exact solution for thetransient one-dimensional temperature and transient thermoelastic response of amultilayered hollow circular disk.
As the numerical example, we carried out numerical calculations forthe functionally graded materials composed of titanium alloy (Ti-6Al-4V) andzirconium oxide (ZrO2) and examined the behaviors in the transient statefor the temperature change, the displacement, the thermal stress displacements.Furthermore, the inuence of the functional grading on the thermal stresses isinvestigated. We obtain the following results.
(1) It is possible to decrease the maximum values of thermal stresses usingthe multilayered FGM model with piecewise power law nonhomogeneity ascompared with the one-layed FGM model.
(2) The values of the temperature, displacement, thermal stress rr decrease whenthe Biots number of the upper and lower surfaces increases.
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