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Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow sphere
Composite Structures 81 (2007) 540–549 www.elsevier.com/locate/compstruct
Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow sphere Yoshihiro Ootao *, Yoshinobu Tanigawa Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University 1-1 Gakuen-cho, Nakaku, Sakai 599-8531, Japan Available online 21 November 2006
Abstract This paper is concerned with the theoretical treatment of transient piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow sphere due to uniform heat supply. The transient one-dimensional temperature is analyzed by the method of Laplace transformation. The thermal, thermoelastic and piezoelectric constants of the hollow sphere are expressed as power functions of the radial coordinate. The one-dimensional solution for the temperature change in a transient state, and piezothermoelastic response of a functionally graded thermopiezoelectric hollow sphere is obtained herein. Some numerical results for the temperature change, displacement, stress and electric potential distributions are shown. Furthermore, the influence of the nonhomogeneity of the material upon the temperature change, displacement, stresses and electric potential is investigated. 2006 Elsevier Ltd. All rights reserved. Keywords: Piezothermoelasticity; Functionally graded material; Hollow sphere; Transient state; One-dimensional problem
1. Introduction Functionally graded materials (FGMs) are made of a mixture with arbitrary composition of two different materials, and the volume fraction of each material changes continuously and gradually. The FGMs concept is applicable to many industrial fields such as aerospace, nuclear energy, chemical plant, electronics, biomaterials and so on. On the other hand, piezoelectric materials have coupled effects between the elastic field and the electric field, and have been widely used as the actuators or sensors in smart composite material systems. Many analytical studies concerning piezoelastic or piezothermoelastic problems have been reported, and their several books have been published [1,2]. Recently a new type of piezoelectric material with material constants varying continuously in the thickness direction, named functionally graded piezoelectric materials, has been developed [3–5]. It is possible to produce large displacements and reduce the stresses when the functionally *
Corresponding author. Tel.: +81 72 254 9210; fax: +81 72 254 9904. E-mail address: [email protected] (Y. Ootao).
0263-8223/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.10.002
graded piezoelectric materials are used as an actuator. Therefore, the piezoelastic or piezothermoelastic problems for these functionally graded piezoelectric materials become important, and there are several analytical studies concerned with these problems. The analytical solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting was obtained in [6]. The elastic analyses of functionally graded piezoelectric plates are reported in [7,8] by using the 2D-elastic model or classical lamination theory. The electromechanical behavior of functionally graded piezoelectric shells was examined in [9] by using a high order theory. The free vibration problem of functionally graded piezoelectric rectangular plate was analyzed in [10] by using a laminated approximation on the basis of three-dimensional theory. On the other hand, the exact treatments for functionally graded piezoelectric materials have been done. The piezoelastic problems of functionally graded piezoelectric spherical shells whose material properties vary with the power product form through the radial direction were treated in [11,12]. The three-dimensional piezoelastic problem of a simply supported functionally graded piezoelectric plate whose
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
material properties vary exponentially in the thickness direction was treated in [13]. The piezoelastic problem of a functionally graded piezoelectric cantilever beam under load was treated in [14]. The 3D Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces were obtained in [15]. The exact solutions of a simply supported functionally graded piezoelectric plate/laminate were obtained in [16,17]. The piezoelastic problem of a functionally graded piezoelectric hollow cylinder, whose modified modulus of elasticity varies linearly in the radial direction, was treated in [18]. The dynamic stability of functionally graded piezoelectric circular cylindrical shells were treated in [19]. As piezothermoelastic problems, a smart functionally graded thermopiezoelectric structures was analyzed in [20] by using finite element method. An exact solution for a functionally graded thermopiezoelectric cylinder shell subjected to axisymmetric thermal or mechanical loading was obtained in [21]. The exact solutions of functionally graded piezothermoelastic cantilevers [22] or simply supported functionally graded piezothermoelastic plates [23] were obtained. These papers, however, treated only the piezothermoelastic problems under the uniform heating or the steady temperature distribution. It is well-known that thermal stress distributions in a transient state can show large values compared with the one in a steady state. Therefore, the transient piezothermoelastic problems become important. The twodimensional solution for transient piezothermoelasticity of a thick functionally graded thermopiezoelectric strip was obtained in [24]. To the author’s knowledge, the exact analysis for a transient piezothermoelastic problem of functionally graded thermopiezoelectric hollow sphere has not been reported. The present paper is concerned with the one-dimensional transient piezothermoelasticity involving a functionally graded thermopiezoelectric hollow sphere whose thermal, thermoelastic and piezoelectric constants are assumed to vary with the power product form of radial coordinate variable. The exact solution for the one-dimensional transient piezothermoelastic problem of the functionally graded thermopiezoelectric hollow sphere is developed. 2. Analysis A functionally graded thermopiezoelectric hollow sphere that has nonhomogeneous thermal, mechanical and piezoelectric properties in the radial direction is considered herein. The hollow sphere’s inner and outer radii are designated ra and rb, respectively. 2.1. Heat conduction problem
and outer surfaces are designated by ha and hb. Then the temperature distribution shows a one-dimensional distribution in r direction, and the transient heat conduction equation is taken in the following form: oT 1 o 2 oT ¼ 2 kðrÞr cðrÞqðrÞ ð1Þ ot r or or The thermal conductivity k and the heat capacity per unit volume cq are assumed to take the following forms: m r kðrÞ ¼ k0 ð2Þ rb k r cðrÞqðrÞ ¼ c0 q0 ð3Þ rb where m and k are nonhomogeneous parameters. Substituting Eqs. (2) and (3) into Eq. (1), the transient heat conduction equation in dimensionless form is oT oT o2 T ¼ ðm þ 2Þrmk1 þ rmk 2 os or or
ð4Þ
The initial and thermal boundary conditions in dimensionless form are s ¼ 0; r ¼ ra ; r ¼ 1;
T ¼0
ð5Þ
oT H a T ¼ H a T a or oT þ H bT ¼ H bT b or
ð6Þ ð7Þ
In Eqs. (1)–(7), the following dimensionless values are introduced: ðT ; T a ; T b Þ ¼ j0 ¼
k0 ; c0 q0
ðT ; T a ; T b Þ ðr; ra Þ ðr; ra Þ ¼ ; T0 rb j0 t s ¼ 2 ; ðH a ; H b Þ ¼ ðha ; hb Þrb rb
ð8Þ
where T is the temperature change, t is time and T0 and j0 are typical values of temperature and thermal diffusivity, respectively. To solve the fundamental Eq. (4), the Laplace transformation with respect to the variable s is introduced. Performing this integral transformation under the condition (5), we obtain d2 T 1 dT p mk T ¼ 0 þ ðm þ 2Þ 2 r dr r dr
ð9Þ
where the symbol (*) means the integral transformation with respect to the variable s, and the parameter of the transformation is denoted by p[ = (2 m + k)2l2/4]. To solve the fundamental Eq. (9) we introduce the following variable and the auxiliary function U as n ¼ rð2mþkÞ=2
It is assumed that the hollow sphere is initially at zero temperature and is suddenly heated from the inner and outer surfaces by surrounding media of temperatures Ta and Tb. The relative heat transfer coefficients on the inner
541
T ¼n
ðmþ1Þ=ð2mþkÞ
ð10Þ U
ð11Þ
Taking into account Eqs. (10) and (11), we obtain the next fundamental equation for U
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d2 U 1 dU c2 2 þ l 2 U ¼ 0 þ n dn dn2 n
ð12Þ
where mþ1 c¼ 2 m þ k
ð13Þ
The solution of Eq. (12) is
U ¼ AJ c ðlnÞ þ BY c ðlnÞ
ð14Þ
From Eqs. (10), (11) and (14), we obtain the next equation for T T ¼ rðmþ1Þ=2 ½AJ c ðlrð2mþkÞ=2 Þ þ BY c ðlrð2mþkÞ=2 Þ
ð15Þ
where Jc( ) and Yc( ) are the Bessel functions of the first and second kind of order c, respectively. A and B are unknown constants, which must be determined from the boundary conditions (6) and (7). Accomplishing the inverse Laplace transformation on Eq. (15), the temperature solution is shown as follows: 1 0 m1 ðA r þ B0 Þ F 2 1 X 2rðmþ1Þ=2 exp½ð2 m þ kÞ l2j s=4 þ lj D0 ðlj Þ j¼1
T ¼
½AJ c ðljrð2mþkÞ=2 Þ þ BY c ðljrð2mþkÞ=2 Þ
e22 ¼ 1
e12 ¼
H a ln ra ;
ð19Þ
e21 ¼ H b ;
if m ¼ 1
c1 ¼ H a T a ;
ð18Þ
ð20Þ
c2 ¼ H b T b
ð21Þ
0
In Eqs. (16) and (17), D (lj) is dD D0 ðlj Þ ¼ dl l¼lj
ð22Þ
and lj represents the jth positive roots of the following transcendental equation: ð23Þ
2.2. Piezothermoelastic problem if m 6¼ 1
2
1 X 2 exp½ðk þ 3Þ l2j s=4 1 0 ðA þ B0 ln rÞ þ F lj D0 ðlj Þ j¼1
½AJ 0 ðljrðkþ3Þ=2 Þ þ BY 0 ðljrðkþ3Þ=2 Þ
e11 ¼ H a ;
r1 a
DðlÞ ¼ 0
ð16Þ T ¼
1 ½cð2 m þ kÞ ðm þ 1Þ þ H b J c ðlÞ 2 lð2 m þ kÞ J cþ1 ðlÞ; 2 1 ½cð2 m þ kÞ ðm þ 1Þ þ H b Y c ðlÞ a22 ¼ 2 lð2 m þ kÞ Y cþ1 ðlÞ 2 e11 ¼ ram1 ½ðm þ 1Þr1 e12 ¼ H a ; a þ H a ; e21 ¼ ðm þ 1Þ þ H b ; e22 ¼ H b if m 6¼ 1;
a21 ¼
if m ¼ 1
ð17Þ
In Eqs. (16) and (17), D and F are the determinants of 2 · 2 matrix [akl] and [ekl], respectively; the coefficients A and B are defined as the determinant of the matrix similar to the coefficient matrix [akl], in which the first column or second column is replaced by the constant vector {ck}, respectively; similarly, the coefficients A0 and B0 are defined as the determinant of the matrix similar to the coefficient matrix [ekl], in which the first column or second column is replaced by the constant vector {ck}, respectively. The elements of the coefficient matrices [akl], [ekl] and the constant vector {ck} are given as follows: 1 a11 ¼ raðmþ1Þ=2 ½cð2 m þ kÞ 2ra ðm þ 1Þ H a gJ c ðlrað2mþkÞ=2 Þ
lð2 m þ kÞ ðmkÞ=2 ra J cþ1 ðlrað2mþkÞ=2 Þ ; 2 1 ½cð2 m þ kÞ a12 ¼ raðmþ1Þ=2 2ra
Consider the transient piezothermoelasticity of a functionally graded hollow thermopiezoelectric sphere as a one-dimensional problem. The displacement–strain relations are expressed in dimensionless form as follows: ur err ¼ ur;r ; ehh ¼ e// ¼ ð24Þ r where a comma denotes partial differentiation with respect to the variable that follows. For the thermopiezoelectric material of crystal class 6 mm, the constitutive relation are expressed in dimensionless form as follows: r T ; r b rr ¼ C 11err þ 2C 12ehh e1 E r h T r b // ¼ C 12err þ ðC 22 þ C 23 Þehh e2 E hh ¼ r r
ð25Þ
where ^r ¼ C 11 ar þ 2C 12 ah ; b ^ ¼ C 12 ar þ ðC 22 þ C 23 Þah b h
ð26Þ
The constitutive equations for the electric field in dimensionless form are given by Dr ¼ e1err þ 2e2ehh þ g1 Er þ p1 T
ð27Þ
The relations between the electric field intensity and the electric potential / in dimensionless form are defined by ;r Er ¼ / ð28Þ
ðm þ 1Þ H a gY c ðlrað2mþkÞ=2 Þ
The equilibrium equation is expressed in dimensionless form as follows:
lð2 m þ kÞ ðmkÞ=2 ra Y cþ1 ðlrað2mþkÞ=2 Þ ; 2
hh Þ=r ¼ 0 rr;r þ 2ð rrr r r
ð29Þ
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
If the electric charge density is absent, the equation of electrostatics is expressed in dimensionless form as follows: Dr;r þ 2Dr =r ¼ 0
ð30Þ
543
rr ¼ 0 r
r ¼ ra ; 1;
ð36Þ
The boundary conditions of inner and outer surfaces for the electric field are expressed by
The elastic stiffness constants Ckl, the piezoelectric coefficients ek, the dielectric constant g1, the coefficients of linear thermal expansion ak and the pyroelectric constant p1 in dimensionless form are assumed to take the following forms
r ¼ ra ;
¼ 0ðboundary condition AÞ /
r ¼ 1;
Dr ¼ 0 /¼0
C kl ¼ C 0klrl ;
The solutions of Eqs. (33) and (34) are assumed in the following form:
ak ¼
a0k rb ;
ek ¼ e0k rl ; p1 ¼
g1 ¼ g01rl ;
p01rbþl
ð31Þ
where l is a nonhomogeneous parameter of the elastic stiffness constant, the piezoelectric coefficient and the dielectric constant. b is a nonhomogeneous parameter of the coefficient of linear thermal expansion. Furthermore, l and b are nonhomogeneous parameters of the pyroelectric constant. In Eqs. (24)–(31), the following dimensionless values are introduced: rkl ekl kl ¼ r ; ekl ¼ ; a0 Y 0 T 0 a0 T 0 ur ðC kl ; C 0kl Þ ; ðC kl ; C 0kl Þ ¼ ; ur ¼ Y0 a0 T 0 r b ðak ; a0k Þ ðek ; e0k Þ ; a0k Þ ¼ ; ðek ; e0k Þ ¼ ð ak ; a0 Y 0 jd 0 j ðg ; g0 Þ ð g1 ; g01 Þ ¼ 1 12 ; Y 0 jd 0 j ðp ; p0 Þ Dr ð p1 ; ; p01 Þ ¼ 1 1 ; Dr ¼ a0 Y 0 jd 0 j a0 Y 0 T 0 jd 0 j ¼ /jd 0 j ; Er ¼ Er jd 0 j / ð32Þ a0 T 0 r b a0 T 0 where rkl are the stress components, ekl are the strain tensor, ur is the displacement in the radial direction, Dr is the electric displacement in the radial direction, and a0, Y0 and d0 are the typical values of the coefficient of linear thermal expansion, Young’s modulus of elasticity and piezoelectric modulus, respectively. Substituting Eqs. (24), (28) and (31) into Eqs. (25) and (27), and later into Eqs. (29) and (30), the governing equations of the displacement and the electric potential in dimensionless form are written as C 011 ½ ur;rr þ ðl þ 2Þur;r =r þ 2½C 012 ðl þ 1Þ C 022 C 023 ur =r2 ;rr þ ½e0 ðl þ 2Þ 2e0 / ;r =r þ e0 / 1
1
e01 ur;rr
þ ½e01 ðl þ 2Þ þ 2e02 ur;r =r þ 2e02 ðl þ 1Þ ur =r2 ;rr þ ðl þ 2Þ/ ;r =r ¼ g01 ½/ p01rb ½T ;r þ ðl þ
ð37Þ
ð38Þ
In Eq. (38), the first term on the right side gives the homogeneous solution and the second term of right side gives the particular solution. Now consider the homogeneous solution, and introduce the following equation: r ¼ exp ðsÞ
ð39Þ
Changing a variable with the use of Eq. (39), the homogeneous expression of Eqs. (33) and (34) are c ¼ 0 ½D2 þ ðl þ 1ÞD ca urc þ ½D2 þ ðl þ 1 cb ÞDU
ð40Þ
2
½D þ ðl þ 1 þ cb ÞD þ cb ðl þ 1Þurc c ¼ 0 cc ½D2 þ ðl þ 1ÞDU
ð41Þ
where d ds 2½C 022 þ C 023 C 012 ðl þ 1Þ ca ¼ ; C 011
ð42Þ
D¼
cc ¼
C 011 g01 2 ðe01 Þ
;
cb ¼
2e02 ; e01
0 c c ¼ e1 / U C 011
ð43Þ
We show uðiÞ rc and Uc as follows: 0 0 ½uðiÞ rc ðqÞ; Uc ðqÞ ¼ ðU rc ; Uc Þ expðksÞ
ð44Þ
Substituting Eq. (44) into Eqs. (40) and (41), the condition that non-trivial solutions of ðU 0rc ; U0c Þ exist leads to the following equation. kðk þ l þ 1Þ½k2 þ ðl þ 1Þk
cb ðcb l 1Þ þ ca cc ¼0 cc þ 1 ð45Þ
ð33Þ We now introduce the following expression: b þ 2ÞT =r
ð34Þ where 0 0 0 0 ¼ C 0 b ah ; r 11 ar þ 2C 12 0 0 0 0 bh ¼ C 12 a0h ar þ ðC 22 þ C 023 Þ
ðboundary condition BÞ;
ur ¼ urc þ urp ; ¼/ c þ / p: /
2
0 ðl þ b þ 2Þ 2b 0 rb1 T þ b 0rb T ;r ¼ ½b r h r
or
ð35Þ
If the inner and outer surfaces are traction free, the boundary conditions of inner and outer surfaces can be represented as follows:
H¼
4½cb ðcb l 1Þ þ ca cc þ ðcc þ 1Þðl þ 1Þ2 cc þ 1
ð46Þ
When H is positive, there are k0 = 0, k1 = (l + 1) and two distinct real roots as follows: pffiffiffiffi pffiffiffiffi ðl þ 1Þ þ H ð1 þ 1Þ H ; k3 ¼ ð47Þ k2 ¼ 2 2 When H is negative, there are k0 = 0, k1 = (l + 1) and one pair of conjugate complex roots as follows:
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Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
kJ ¼ aJ jbJ
ð48Þ
For example, when H is positive and l 5 1, urc ðrÞ and c ðrÞ of the homogeneous solution are given by the / following expressions:
ð49Þ
urc ðrÞ ¼
where lþ1 ; aJ ¼ 2
pffiffiffiffiffiffiffiffi H bJ ¼ 2
3 X
F J rkJ ;
J ¼1
Case 1. Real roots for k (k 5 0, l 5 1)
0
Given JN real roots for k except for k0 = 0 if l 5 1, urc ðrÞ c ðrÞ are given by the following expressions: and U urc ðrÞ ¼
JN X
F J rkJ ;
J ¼1
c ðrÞ ¼ U
JN X
M J ðkJ ÞF J rkJ
ð50Þ
J ¼1
3 X
M J ðkJ ÞF J rkJ þ G0
ð56Þ
J ¼1
In order to obtain the particular solution, the series expansions of the Bessel functions are used as shown in Appendix A. Eqs. (16) and (17) can be written as the following expression: T¼
1 X
fa1n ðsÞrnð2mþkÞ þ b1n ðsÞrnð2mþkÞðmþ1Þ g
n¼0
where M J ðkJ Þ ¼
c ðrÞ ¼ C 11 / e01
mþ1 6¼ integer 2mþk 1 X T¼ fa2n ðsÞrnð2mþkÞ þ c2n ðsÞrnð2mþkÞ lnrg if
k2J
þ ðl þ 1ÞkJ ca kJ ðkJ þ l þ 1 cb Þ
ð51Þ
n¼0 c1 X
In Eq. (50), FJ are unknown constants.
þ
Case 2. For k0 = 0 (l 5 1)
n¼0
c ðrÞ are given by the folFor real root k0 ¼ 0; urc ðrÞ and U lowing expressions:
T¼
1 X
fa3n ðsÞrnð2mþkÞðmþ1Þ þ c3n ðsÞrnð2mþkÞðmþ1Þ lnrg
n¼0 c1 X
ð52Þ
mþ1 ¼ positive integer 2mþk ð58Þ
urc ðrÞ ¼ 0; 0 c ðrÞ ¼ e1 G0 U C 011
b2n ðsÞrnð2mþkÞðmþ1Þ if
ð57Þ
þ
b3n ðsÞrnð2mþkÞ if
n¼0 1 X
mþ1 ¼ negative integer 2mþk
fa4n ðsÞrnðkþ3Þ þ c4n ðsÞrnðkþ3Þ lnrg if m ¼ 1
ð59Þ
In Eq. (52), G0 is unknown constant.
T¼
Case 3. For k0 = k1 = 0 (l = 1)
Expressions for the functions ain(s), bin(s) and cin(s) in Eqs. (57)–(60) are omitted here for the sake of brevity. For example, when n(2 m + k) + b + 150 and n(2 m + k) + b m50, the particular solutions urp and p are assumed as the following forms: / 1 X urp ¼ ff1an ðsÞrnð2mþkÞþbþ1 þ f1bn ðsÞrnð2mþkÞmþb g;
Eq. (45) has double root if l = 1. For double root c ðrÞ are given by the following k0 ¼ k1 ¼ 0; urc ðrÞ and U expressions: urc ðrÞ ¼ F 0 ; 0 c ðrÞ ¼ e1 G0 ca F 0 ln r U 0 cb C 11
ð53Þ
1 X
fh1an ðsÞrnð2mþkÞþbþ1 þ h1bn ðsÞrnð2mþkÞmþb g
n¼0
Case 4. Complex roots for k
mþ1 6¼ integer ð61Þ 2mþk 1 X urp ¼ ff2an ðsÞrnð2mþkÞþbþ1 þ f2cn ðsÞrnð2mþkÞþbþ1 ln rg if
c ðrÞ are For one pair of complex roots for k, urc ðrÞ and U given by the following expressions:
n¼0 c1 X
þ ð54Þ
where CJ ¼ Re½M J ðkJ ÞjkJ ¼aJ þjbJ ; XJ ¼ Im½M J ðkJ ÞjkJ ¼aJ þjbJ
n¼0
p ¼ /
In Eq. (53), F0 and G0 are unknown constants.
urc ðrÞ ¼ C 1J raJ cosðbJ ln rÞ þ C 2J raJ sinðbJ ln rÞ; c ðrÞ ¼ C 1J raJ ½CJ cosðbJ ln rÞ XJ sinðbJ ln rÞ U C 2J raJ ½XJ cosðbJ ln rÞ þ CJ sin ðbJ ln rÞ
ð60Þ
n¼0
f2bn ðsÞrnð2mþkÞmþb ;
n¼0
p ¼ /
1 X
fh2an ðsÞrnð2mþkÞþbþ1 þ h2cn ðsÞrnð2mþkÞþbþ1 ln rg
n¼0
ð55Þ
In Eqs. and (55), j, Re[ ] and Im[ ] are imaginary unit pffiffiffiffiffiffi(48) ffi j ¼ 1, real part and imaginary part, respectively. Furthermore, in Eq. (54), C1J and C2J are unknown constants.
þ
c1 X
h2bn ðsÞrnð2mþkÞmþb
n¼0
if
mþ1 ¼ positive integer 2mþk
ð62Þ
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
urp ¼
1 X
ff3an ðsÞrnð2mþkÞmþb þ f3cn ðsÞrnð2mþkÞmþb ln rg
Table 1 Nonhomogeneous parameters
n¼0
þ
c1 X
m
f3bn ðsÞrnð2mþkÞþbþ1 ;
1 X fh3an ðsÞrnð2mþkÞmþb þ h3cn ðsÞrnð2mþkÞmþb ln rg n¼0
þ
c1 X
mþ1 ¼ negative integer if 2mþk 1 X rp ¼ u ff4an ðsÞrnðkþ3Þþbþ1 þ f4cn ðsÞrnðkþ3Þþbþ1 ln rg; p ¼ /
l
0 0 0
0 0 0
0.01 0.01 0.01
Material 2
0 0 0
1 0 1
0 0 0
0.01 0.01 0.01
Material 3
0 0 0
0 0 0
1 0 1
0.01 0.01 0.01
Material 4
0 0 0
0 0 0
0 0 0
1 0.01 1
ð63Þ
n¼0 1 X
b
1 0 1
h3bn ðsÞrnð2mþkÞþbþ1
n¼0
k
Material 1
n¼0
p ¼ /
545
fh4an ðsÞrnðkþ3Þþbþ1 þ h4cn ðsÞrnðkþ3Þþbþ1 ln rg
n¼0
if m ¼ 1
ð64Þ
where fian(s), fibn(s), ficn(s), hian (s), hibn(s) and hicn(s) can be obtained from Eqs. (33) and (34). The particular solutions p for n(2 m + k) + b + 1 = 0 or n(2 m + k) + urp and / b m = 0 are omitted here for the sake of brevity. Then, the stress components and the electric displacement can be evaluated by substituting Eq. (38) into Eqs. (24) and (28), and later into Eqs. (25) and (27). The unknown constants in Eq. (56) are determined so as to satisfy the boundary conditions (36) and (37). 3. Numerical results To illustrate the foregoing analysis, numerical parameters of heat conduction and shape are presented as follows: Ha = Hb = 1.0, T a ¼ 0;
T b ¼ 1;
ra ¼ 0:7
ð65Þ
The hollow sphere is heated from the outer surface. The material constants are taken with reference to cadmium selenide as follows: a0r ¼ 2:458 106 1=K;
k is a nonhomogeneous parameter of the heat capacity per unit volume, b is a nonhomogeneous parameter of the coefficient of linear thermal expansion, and l is a nonhomogeneous parameter of the elastic stiffness constant, the piezoelectric coefficient, and the dielectric constant, respectively. Values given for material 1 correspond to the results for nonhomogeneity of the thermal conductivity; values given for material 2 correspond to the results for nonhomogeneity of the heat capacity per unit volume; values given for material 3 correspond to the results for nonhomogeneity of the coefficient of linear thermal expansion; and values given for material 4 correspond to the results for nonhomogeneity of the elastic stiffness constant, the piezoelectric coefficients, and the dielectric constant. When the nonhomogeneous parameters equal zero, the material constants are constant. As the derived solution breaks down if l = 0, the nonhomogeneous parameter l is taken to be 0.01. In materials 1–4 listed in Table 1, the second values are closely analogous to those for homogeneous material. Figs. 1–10 show the numerical results for the boundary condition A. The results for material 1 are shown in Figs. 1–4. Fig. 1 shows the variation of temperature change in
a0h ¼ 4:396 106 1=K;
C 011 ¼ 83:6 GPa; C 022 ¼ 74:1 GPa; C 012 ¼ 39:3 GPa; C 023 ¼ 45:2 GPa;
e02 ¼ 0:16 C=m2 ;
e01 ¼ 0:347C=m2 ; g01 ¼ 9:03 1011 C2 =Nm2 ;
p01 ¼ 2:94 106 C=m2 K;
kr0 ¼ 12:9 W=mK; c0 ¼ 0:4614 103 J=kg K; q0 ¼ 5:684 103 kg=m3
ð66Þ
The typical values of material properties such as a0, Y0 and d0, used to normalize the numerical data, are a0 ¼ a0h ;
Y 0 ¼ 42:8 GPa;d 0 ¼ 3:92 1012 C=N
ð67Þ
The nonhomogeneous parameters adopted for the numerical calculations are shown in Table 1. In Table 1, m is a nonhomogeneous parameter of the thermal conductivity,
Fig. 1. Variation of temperature change in the radial direction (boundary condition A and material 1).
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Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
Fig. 2. Variation of displacement ur in the radial direction (boundary condition A and material 1).
Fig. 4. Variation of electric potential in the radial direction (boundary condition A and material 1).
the radial direction. As shown in Fig. 1, the temperature rises as the time proceeds and is greatest in a steady state. From Fig. 1, it can be seen that the temperature change on the heated surface increases when the parameter m increases. Fig. 2 shows the variation of displacement ur in the radial direction. It can be seen from Fig. 2 that the displacement ur increases when the parameter m increases. rr and r hh shown in The variations of normal stresses r
Fig. 5. Variation of temperature change in the radial direction (boundary condition A and material 2).
Fig. 3. Variation of thermal stresses in the radial direction (boundary hh . condition A and material 1): (a) normal stress r rr and (b) normal stress r
Fig. 3a and b, respectively. From Fig. 3a, the maximum tensile stress occurs in a steady state inside the hollow sphere without distinction of parameter m. From Fig. 3b, the large compressive stress occurs on the heated surface and the tensile stress occurs on the inner surface without distinction of parameter m. From Fig. 3, the absolute maximum values of the stresses increases when the parameter m increases. Fig. 4 shows the variation of electric potential / in the radial direction. As shown in Fig. 4, the electric potential rises as the time proceeds and is greatest in a steady state. It can be seen that the electric potential increases when the parameter m increases. The results for material 2 are shown in Figs. 5–7. Fig. 5 shows the variation of temperature change in the radial rr and r hh direction. The variations of normal stresses r are shown in Fig. 6a and b, respectively. Fig. 7 shows the in the radial direction. It variation of electric potential / can be seen from Figs. 5–7 that the distributions of the temperature change, thermal stresses and electric potential in a transient state are changed when the parameter k changes, while their distributions in a steady state are same without distinction of parameter k. From Figs. 5–7, the absolute values of the temperature change and electric potential in
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
547
Fig. 6. Variation of thermal stresses in the radial direction (boundary rr and (b) normal stress r hh . condition A and material 2): (a) normal stress r
Fig. 8. Variation of thermal stresses in the radial direction(boundary rr and (b) normal stress r hh . condition A and material 3): (a) normal stress r
Fig. 7. Variation of electric potential in the radial direction (boundary condition A and material 2).
Fig. 9. Variation of electric potential in the radial direction (boundary condition A and material 3).
a transient state increase when the parameter k increases, rr and r hh in a while the absolute values of the stresses r transient state decrease slightly when the parameter k increases. The results for material 3 are shown in Figs. 8 and 9. rr and r hh are shown The variations of normal stresses r in Fig. 8a and b, respectively. Fig. 9 shows the variation in the radial direction. From of electric potential / rr and r hh Fig. 8, the absolute value of the stresses r increases when the parameter b increases. From Fig. 9,
the electric potential increases when the parameter b decreases. The results for material 4 are shown in Fig. 10. The varihh are shown in Fig. 10a rr and r ations of normal stresses r and b, respectively. From Fig. 10, the absolute values of the stresses increase when the parameter l decreases. Fig. 11 shows the numerical result for the boundary condition B. The variation of electric potential in the radial direction for material 1 is shown in Fig. 11. From Fig. 11, it can be seen that the positive maximum electric
548
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549 Table 2 Comparison between the numerical results for boundary condition A and those for boundary condition B in a steady state (material 1) s=1
m
Boundary condition ¼ 0Þ Aðr ¼ ra ; /
Bðr ¼ ra ; Dr ¼ 0Þ
ur ðr ¼ ra Þ
1 0 1
0.5594 0.5069 0.4484
0.5316 0.4817 0.4262
ur ðr ¼ 1Þ
1 0 1
0.6848 0.6206 0.5493
0.6711 0.6083 0.5384
rr ðr ¼ 0:85Þ r
1 0 1
0.03471 0.03454 0.03386
0.03332 0.03328 0.03275
hh ðr ¼ ra Þ r
1 0 1
0.2552 0.2452 0.2314
0.2442 0.2353 0.2226
hh ðr ¼ 1Þ r
1 0 1
0.1600 0.1635 0.1658
0.1539 0.1581 0.1610
Table 3 Comparison between the numerical results for boundary condition A and those for boundary condition B in a transient state (material 1) s = 0.05
m
Boundary condition ¼ 0Þ Aðr ¼ ra ; /
Fig. 10. Variation of thermal stresses in the radial direction (boundary rr and (b) normal stress r hh . condition A and material 4): (a) normal stress r
ur ðr ¼ ra Þ
1 0 1
0.1396 0.1389 0.1375
0.1329 0.1323 0.1310
potential occurs in the inner surface, and the value rises as the time proceeds. The electric potential increases when the parameter m increases as in the case with boundary condition A. In order to examine the influence of the electric boundary condition, the numerical results for boundary condition A and boundary condition B are shown in Tables 2 and 3, which show typical values of the displacement and stresses for material 1. Tables 2 and 3 show the numerical results in
ur ðr ¼ 1Þ
1 0 1
0.1721 0.1712 0.1693
0.1688 0.1679 0.1661
rr ðr ¼ 0:85Þ r
1 0 1
0.03000 0.02739 0.02528
0.02963 0.02705 0.02495
r hh ðr ¼ ra Þ
1 0 1
0.1562 0.1422 0.1312
0.1534 0.1396 0.1286
r hh ðr ¼ 1Þ
1 0 1
0.1768 0.1673 0.1593
0.1754 0.1659 0.1578
Bðr ¼ ra ; Dr ¼ 0Þ
a steady state and a transient state (s = 0.05), respectively. Tables 2 and 3 show that the absolute values of displacement and stresses for boundary condition A are slightly larger than those for boundary condition B. 4. Conclusion
Fig. 11. Variation of electric potential in the radial direction (boundary condition B and material 1).
In the present article, the piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow sphere that has nonhomogeneous thermal and mechanical properties in the radial direction is analyzed. Thermal conductivity, the heat capacity per unit volume, the elastic stiff-
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
ness constants, the piezoelectric coefficients, the dielectric constant, the coefficient of linear thermal expansion and the pyroelectric constant are expressed as power functions of the radial coordinate. The exact solution for the transient one-dimensional temperature and transient piezothermoelastic response of a functionally graded thermopiezoelectric hollow sphere is obtained. The present solution can serve as a benchmark to the analysis of functionally graded thermopiezoelectric hollow sphere based on various numerical methods. Appendix A. Bessel functions
J c ðxÞ ¼
n 2nþc 1 X ð1Þ ðx=2Þ n!Cðc þ n þ 1Þ n¼0
ðA:1Þ
1 ½cos cp J c ðxÞ J c ðxÞ if c 6¼ integer; sin cp
2 x Y c ðxÞ ¼ J c ðxÞ c þ ln p 2 ! cþn n 1 n X 1 ð1Þ x 2nþc X 1 X 1 þ p n¼0 n!ðc þ nÞ! 2 M M¼1 M M¼1 Y c ðxÞ ¼
c1 1 X ðc n 1Þ! x 2nc p n¼0 n! 2
if c ¼ integer ðA:2Þ
where c* is Euler’s constant. References [1] Qin Q-H. Fracture mechanics of piezoelectric materials. Southampton: WIT; 2001. [2] Ding HJ, Chen WQ. Three dimensional problems of piezoelasticity. New York: Nova Science Publishers; 2001. [3] Wu CCM, Kahn M, Moy W. Piezoelectric ceramics with functionally gradients: a new application in material design. J Am Ceram Soc 1996;79:809–12. [4] Shelley WF, Wan S, Bowman KJ. Functionally graded piezoelectric ceramics. Mater Sci Forum 1999;308-311:515–20. [5] Zhu XH, Zu J, Meng ZY, Zhu JM, Zhou SH, Li Q, et al. Microdisplacement characteristics and microstructures of functionally graded piezoelectric ceramic actuator. Mater Des 2000;21:561–6. [6] Lim CW, He LH. Exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting. Int J Mech Sci 2001;43:2479–92.
549
[7] Almajid A, Taya M. 2D-elastic analysis of FGM piezo-laminates under cylindrical bending. J Intelligent Mater Sys Struct 2001;12: 341–51. [8] Almajid A, Taya M, Hudnut S. Analysis of out-of plane displacement and stress field in a piezocomposite plate with functionally graded microstructure. Int J Solids Struct 2001;38:3377–91. [9] Wu XH, Chen C, Shen YP, Tian XG. A high order theory for functionally graded piezoelectric shells. Int J Solids Struct 2002;39: 5325–44. [10] Chen WQ, Ding HJ. On free vibration of a functionally graded piezoelectric rectangular plate. Acta Mech 2002;153:207–16. [11] Chen WQ, Ding HJ, Liang J. The exact elasto-electric field of a rotating piezoceramic spherical shell with a functionally gradient property. Int J Solids Struct 2001;38:7015–27. [12] Chen WQ, Lu Y, Ye GR, Cai JB. 3D electroelastic fields in a functionally graded piezoceramic hollow sphere under mechanical and electric loading. Arch Appl Mech 2002;72:39–51. [13] Zhong Z, Shang ET. Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate. Int J Solids Struct 2003;40:5335–52. [14] Shi ZF, Chen Y. Functionally graded piezoelectric cantilever beam under load. Arch Appl Mech 2004;74:237–47. [15] Pan E, Han F. Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int J Solids Struct 2005;42:3207–33. [16] Lu P, Lee HP, Lu C. An exact solution for functionally graded piezoelectric laminates in cylinderical bending. Int J Mech Sci 2005;47:437–58. [17] Lu P, Lee HP, Lu C. Exact solutions for simply supported functionally graded piezoelectric laminates by Stroh-like formalism. Compos Struct 2006;72:352–63. [18] Zhang TT, Shi ZF. Elastic solutions of graded piezoelectric hollow cylinders. Key Eng Mater 2006;302-303:658–68. [19] Zhu JQ, Chen C, Shen YP, Wang SL. Dynamic stability of functionally graded piezoelectric circular cylindrical shells. Mater Lett 2005;59:477–85. [20] Wang BL, Noda N. Design of a smart functionally graded thermopiezoelectric composite structure. Smart Mater Struct 2001;10: 189–93. [21] Wu XH, Shen YP, Chen C. An exact solution for functionally graded piezothermoelastic cylindrical shell as sensors or actuators. Mater Lett 2003;57:3532–42. [22] Ying C, Zhifei S. Exact solutions of functionally gradient piezothermoelastic cantilevers and parameter identification. J Intelligent Mater Sys Struct 2005;16:531–9. [23] Zhong Z, Shang E. Exact analysis of simply supported functionally graded piezothermoelectric plates. J Intelligent Mater Sys Struct 2005;16:643–51. [24] Ootao Y, Tanigawa Y. The transient piezothermoelastic problem of a thick functionally graded thermopiezoelectric strip due to nonuniform heat supply. Arch Appl Mech 2005;74:449–65.
Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow sphere Yoshihiro Ootao *, Yoshinobu Tanigawa Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University 1-1 Gakuen-cho, Nakaku, Sakai 599-8531, Japan Available online 21 November 2006
Abstract This paper is concerned with the theoretical treatment of transient piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow sphere due to uniform heat supply. The transient one-dimensional temperature is analyzed by the method of Laplace transformation. The thermal, thermoelastic and piezoelectric constants of the hollow sphere are expressed as power functions of the radial coordinate. The one-dimensional solution for the temperature change in a transient state, and piezothermoelastic response of a functionally graded thermopiezoelectric hollow sphere is obtained herein. Some numerical results for the temperature change, displacement, stress and electric potential distributions are shown. Furthermore, the influence of the nonhomogeneity of the material upon the temperature change, displacement, stresses and electric potential is investigated. 2006 Elsevier Ltd. All rights reserved. Keywords: Piezothermoelasticity; Functionally graded material; Hollow sphere; Transient state; One-dimensional problem
1. Introduction Functionally graded materials (FGMs) are made of a mixture with arbitrary composition of two different materials, and the volume fraction of each material changes continuously and gradually. The FGMs concept is applicable to many industrial fields such as aerospace, nuclear energy, chemical plant, electronics, biomaterials and so on. On the other hand, piezoelectric materials have coupled effects between the elastic field and the electric field, and have been widely used as the actuators or sensors in smart composite material systems. Many analytical studies concerning piezoelastic or piezothermoelastic problems have been reported, and their several books have been published [1,2]. Recently a new type of piezoelectric material with material constants varying continuously in the thickness direction, named functionally graded piezoelectric materials, has been developed [3–5]. It is possible to produce large displacements and reduce the stresses when the functionally *
Corresponding author. Tel.: +81 72 254 9210; fax: +81 72 254 9904. E-mail address: [email protected] (Y. Ootao).
0263-8223/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.10.002
graded piezoelectric materials are used as an actuator. Therefore, the piezoelastic or piezothermoelastic problems for these functionally graded piezoelectric materials become important, and there are several analytical studies concerned with these problems. The analytical solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting was obtained in [6]. The elastic analyses of functionally graded piezoelectric plates are reported in [7,8] by using the 2D-elastic model or classical lamination theory. The electromechanical behavior of functionally graded piezoelectric shells was examined in [9] by using a high order theory. The free vibration problem of functionally graded piezoelectric rectangular plate was analyzed in [10] by using a laminated approximation on the basis of three-dimensional theory. On the other hand, the exact treatments for functionally graded piezoelectric materials have been done. The piezoelastic problems of functionally graded piezoelectric spherical shells whose material properties vary with the power product form through the radial direction were treated in [11,12]. The three-dimensional piezoelastic problem of a simply supported functionally graded piezoelectric plate whose
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
material properties vary exponentially in the thickness direction was treated in [13]. The piezoelastic problem of a functionally graded piezoelectric cantilever beam under load was treated in [14]. The 3D Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces were obtained in [15]. The exact solutions of a simply supported functionally graded piezoelectric plate/laminate were obtained in [16,17]. The piezoelastic problem of a functionally graded piezoelectric hollow cylinder, whose modified modulus of elasticity varies linearly in the radial direction, was treated in [18]. The dynamic stability of functionally graded piezoelectric circular cylindrical shells were treated in [19]. As piezothermoelastic problems, a smart functionally graded thermopiezoelectric structures was analyzed in [20] by using finite element method. An exact solution for a functionally graded thermopiezoelectric cylinder shell subjected to axisymmetric thermal or mechanical loading was obtained in [21]. The exact solutions of functionally graded piezothermoelastic cantilevers [22] or simply supported functionally graded piezothermoelastic plates [23] were obtained. These papers, however, treated only the piezothermoelastic problems under the uniform heating or the steady temperature distribution. It is well-known that thermal stress distributions in a transient state can show large values compared with the one in a steady state. Therefore, the transient piezothermoelastic problems become important. The twodimensional solution for transient piezothermoelasticity of a thick functionally graded thermopiezoelectric strip was obtained in [24]. To the author’s knowledge, the exact analysis for a transient piezothermoelastic problem of functionally graded thermopiezoelectric hollow sphere has not been reported. The present paper is concerned with the one-dimensional transient piezothermoelasticity involving a functionally graded thermopiezoelectric hollow sphere whose thermal, thermoelastic and piezoelectric constants are assumed to vary with the power product form of radial coordinate variable. The exact solution for the one-dimensional transient piezothermoelastic problem of the functionally graded thermopiezoelectric hollow sphere is developed. 2. Analysis A functionally graded thermopiezoelectric hollow sphere that has nonhomogeneous thermal, mechanical and piezoelectric properties in the radial direction is considered herein. The hollow sphere’s inner and outer radii are designated ra and rb, respectively. 2.1. Heat conduction problem
and outer surfaces are designated by ha and hb. Then the temperature distribution shows a one-dimensional distribution in r direction, and the transient heat conduction equation is taken in the following form: oT 1 o 2 oT ¼ 2 kðrÞr cðrÞqðrÞ ð1Þ ot r or or The thermal conductivity k and the heat capacity per unit volume cq are assumed to take the following forms: m r kðrÞ ¼ k0 ð2Þ rb k r cðrÞqðrÞ ¼ c0 q0 ð3Þ rb where m and k are nonhomogeneous parameters. Substituting Eqs. (2) and (3) into Eq. (1), the transient heat conduction equation in dimensionless form is oT oT o2 T ¼ ðm þ 2Þrmk1 þ rmk 2 os or or
ð4Þ
The initial and thermal boundary conditions in dimensionless form are s ¼ 0; r ¼ ra ; r ¼ 1;
T ¼0
ð5Þ
oT H a T ¼ H a T a or oT þ H bT ¼ H bT b or
ð6Þ ð7Þ
In Eqs. (1)–(7), the following dimensionless values are introduced: ðT ; T a ; T b Þ ¼ j0 ¼
k0 ; c0 q0
ðT ; T a ; T b Þ ðr; ra Þ ðr; ra Þ ¼ ; T0 rb j0 t s ¼ 2 ; ðH a ; H b Þ ¼ ðha ; hb Þrb rb
ð8Þ
where T is the temperature change, t is time and T0 and j0 are typical values of temperature and thermal diffusivity, respectively. To solve the fundamental Eq. (4), the Laplace transformation with respect to the variable s is introduced. Performing this integral transformation under the condition (5), we obtain d2 T 1 dT p mk T ¼ 0 þ ðm þ 2Þ 2 r dr r dr
ð9Þ
where the symbol (*) means the integral transformation with respect to the variable s, and the parameter of the transformation is denoted by p[ = (2 m + k)2l2/4]. To solve the fundamental Eq. (9) we introduce the following variable and the auxiliary function U as n ¼ rð2mþkÞ=2
It is assumed that the hollow sphere is initially at zero temperature and is suddenly heated from the inner and outer surfaces by surrounding media of temperatures Ta and Tb. The relative heat transfer coefficients on the inner
541
T ¼n
ðmþ1Þ=ð2mþkÞ
ð10Þ U
ð11Þ
Taking into account Eqs. (10) and (11), we obtain the next fundamental equation for U
542
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
d2 U 1 dU c2 2 þ l 2 U ¼ 0 þ n dn dn2 n
ð12Þ
where mþ1 c¼ 2 m þ k
ð13Þ
The solution of Eq. (12) is
U ¼ AJ c ðlnÞ þ BY c ðlnÞ
ð14Þ
From Eqs. (10), (11) and (14), we obtain the next equation for T T ¼ rðmþ1Þ=2 ½AJ c ðlrð2mþkÞ=2 Þ þ BY c ðlrð2mþkÞ=2 Þ
ð15Þ
where Jc( ) and Yc( ) are the Bessel functions of the first and second kind of order c, respectively. A and B are unknown constants, which must be determined from the boundary conditions (6) and (7). Accomplishing the inverse Laplace transformation on Eq. (15), the temperature solution is shown as follows: 1 0 m1 ðA r þ B0 Þ F 2 1 X 2rðmþ1Þ=2 exp½ð2 m þ kÞ l2j s=4 þ lj D0 ðlj Þ j¼1
T ¼
½AJ c ðljrð2mþkÞ=2 Þ þ BY c ðljrð2mþkÞ=2 Þ
e22 ¼ 1
e12 ¼
H a ln ra ;
ð19Þ
e21 ¼ H b ;
if m ¼ 1
c1 ¼ H a T a ;
ð18Þ
ð20Þ
c2 ¼ H b T b
ð21Þ
0
In Eqs. (16) and (17), D (lj) is dD D0 ðlj Þ ¼ dl l¼lj
ð22Þ
and lj represents the jth positive roots of the following transcendental equation: ð23Þ
2.2. Piezothermoelastic problem if m 6¼ 1
2
1 X 2 exp½ðk þ 3Þ l2j s=4 1 0 ðA þ B0 ln rÞ þ F lj D0 ðlj Þ j¼1
½AJ 0 ðljrðkþ3Þ=2 Þ þ BY 0 ðljrðkþ3Þ=2 Þ
e11 ¼ H a ;
r1 a
DðlÞ ¼ 0
ð16Þ T ¼
1 ½cð2 m þ kÞ ðm þ 1Þ þ H b J c ðlÞ 2 lð2 m þ kÞ J cþ1 ðlÞ; 2 1 ½cð2 m þ kÞ ðm þ 1Þ þ H b Y c ðlÞ a22 ¼ 2 lð2 m þ kÞ Y cþ1 ðlÞ 2 e11 ¼ ram1 ½ðm þ 1Þr1 e12 ¼ H a ; a þ H a ; e21 ¼ ðm þ 1Þ þ H b ; e22 ¼ H b if m 6¼ 1;
a21 ¼
if m ¼ 1
ð17Þ
In Eqs. (16) and (17), D and F are the determinants of 2 · 2 matrix [akl] and [ekl], respectively; the coefficients A and B are defined as the determinant of the matrix similar to the coefficient matrix [akl], in which the first column or second column is replaced by the constant vector {ck}, respectively; similarly, the coefficients A0 and B0 are defined as the determinant of the matrix similar to the coefficient matrix [ekl], in which the first column or second column is replaced by the constant vector {ck}, respectively. The elements of the coefficient matrices [akl], [ekl] and the constant vector {ck} are given as follows: 1 a11 ¼ raðmþ1Þ=2 ½cð2 m þ kÞ 2ra ðm þ 1Þ H a gJ c ðlrað2mþkÞ=2 Þ
lð2 m þ kÞ ðmkÞ=2 ra J cþ1 ðlrað2mþkÞ=2 Þ ; 2 1 ½cð2 m þ kÞ a12 ¼ raðmþ1Þ=2 2ra
Consider the transient piezothermoelasticity of a functionally graded hollow thermopiezoelectric sphere as a one-dimensional problem. The displacement–strain relations are expressed in dimensionless form as follows: ur err ¼ ur;r ; ehh ¼ e// ¼ ð24Þ r where a comma denotes partial differentiation with respect to the variable that follows. For the thermopiezoelectric material of crystal class 6 mm, the constitutive relation are expressed in dimensionless form as follows: r T ; r b rr ¼ C 11err þ 2C 12ehh e1 E r h T r b // ¼ C 12err þ ðC 22 þ C 23 Þehh e2 E hh ¼ r r
ð25Þ
where ^r ¼ C 11 ar þ 2C 12 ah ; b ^ ¼ C 12 ar þ ðC 22 þ C 23 Þah b h
ð26Þ
The constitutive equations for the electric field in dimensionless form are given by Dr ¼ e1err þ 2e2ehh þ g1 Er þ p1 T
ð27Þ
The relations between the electric field intensity and the electric potential / in dimensionless form are defined by ;r Er ¼ / ð28Þ
ðm þ 1Þ H a gY c ðlrað2mþkÞ=2 Þ
The equilibrium equation is expressed in dimensionless form as follows:
lð2 m þ kÞ ðmkÞ=2 ra Y cþ1 ðlrað2mþkÞ=2 Þ ; 2
hh Þ=r ¼ 0 rr;r þ 2ð rrr r r
ð29Þ
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
If the electric charge density is absent, the equation of electrostatics is expressed in dimensionless form as follows: Dr;r þ 2Dr =r ¼ 0
ð30Þ
543
rr ¼ 0 r
r ¼ ra ; 1;
ð36Þ
The boundary conditions of inner and outer surfaces for the electric field are expressed by
The elastic stiffness constants Ckl, the piezoelectric coefficients ek, the dielectric constant g1, the coefficients of linear thermal expansion ak and the pyroelectric constant p1 in dimensionless form are assumed to take the following forms
r ¼ ra ;
¼ 0ðboundary condition AÞ /
r ¼ 1;
Dr ¼ 0 /¼0
C kl ¼ C 0klrl ;
The solutions of Eqs. (33) and (34) are assumed in the following form:
ak ¼
a0k rb ;
ek ¼ e0k rl ; p1 ¼
g1 ¼ g01rl ;
p01rbþl
ð31Þ
where l is a nonhomogeneous parameter of the elastic stiffness constant, the piezoelectric coefficient and the dielectric constant. b is a nonhomogeneous parameter of the coefficient of linear thermal expansion. Furthermore, l and b are nonhomogeneous parameters of the pyroelectric constant. In Eqs. (24)–(31), the following dimensionless values are introduced: rkl ekl kl ¼ r ; ekl ¼ ; a0 Y 0 T 0 a0 T 0 ur ðC kl ; C 0kl Þ ; ðC kl ; C 0kl Þ ¼ ; ur ¼ Y0 a0 T 0 r b ðak ; a0k Þ ðek ; e0k Þ ; a0k Þ ¼ ; ðek ; e0k Þ ¼ ð ak ; a0 Y 0 jd 0 j ðg ; g0 Þ ð g1 ; g01 Þ ¼ 1 12 ; Y 0 jd 0 j ðp ; p0 Þ Dr ð p1 ; ; p01 Þ ¼ 1 1 ; Dr ¼ a0 Y 0 jd 0 j a0 Y 0 T 0 jd 0 j ¼ /jd 0 j ; Er ¼ Er jd 0 j / ð32Þ a0 T 0 r b a0 T 0 where rkl are the stress components, ekl are the strain tensor, ur is the displacement in the radial direction, Dr is the electric displacement in the radial direction, and a0, Y0 and d0 are the typical values of the coefficient of linear thermal expansion, Young’s modulus of elasticity and piezoelectric modulus, respectively. Substituting Eqs. (24), (28) and (31) into Eqs. (25) and (27), and later into Eqs. (29) and (30), the governing equations of the displacement and the electric potential in dimensionless form are written as C 011 ½ ur;rr þ ðl þ 2Þur;r =r þ 2½C 012 ðl þ 1Þ C 022 C 023 ur =r2 ;rr þ ½e0 ðl þ 2Þ 2e0 / ;r =r þ e0 / 1
1
e01 ur;rr
þ ½e01 ðl þ 2Þ þ 2e02 ur;r =r þ 2e02 ðl þ 1Þ ur =r2 ;rr þ ðl þ 2Þ/ ;r =r ¼ g01 ½/ p01rb ½T ;r þ ðl þ
ð37Þ
ð38Þ
In Eq. (38), the first term on the right side gives the homogeneous solution and the second term of right side gives the particular solution. Now consider the homogeneous solution, and introduce the following equation: r ¼ exp ðsÞ
ð39Þ
Changing a variable with the use of Eq. (39), the homogeneous expression of Eqs. (33) and (34) are c ¼ 0 ½D2 þ ðl þ 1ÞD ca urc þ ½D2 þ ðl þ 1 cb ÞDU
ð40Þ
2
½D þ ðl þ 1 þ cb ÞD þ cb ðl þ 1Þurc c ¼ 0 cc ½D2 þ ðl þ 1ÞDU
ð41Þ
where d ds 2½C 022 þ C 023 C 012 ðl þ 1Þ ca ¼ ; C 011
ð42Þ
D¼
cc ¼
C 011 g01 2 ðe01 Þ
;
cb ¼
2e02 ; e01
0 c c ¼ e1 / U C 011
ð43Þ
We show uðiÞ rc and Uc as follows: 0 0 ½uðiÞ rc ðqÞ; Uc ðqÞ ¼ ðU rc ; Uc Þ expðksÞ
ð44Þ
Substituting Eq. (44) into Eqs. (40) and (41), the condition that non-trivial solutions of ðU 0rc ; U0c Þ exist leads to the following equation. kðk þ l þ 1Þ½k2 þ ðl þ 1Þk
cb ðcb l 1Þ þ ca cc ¼0 cc þ 1 ð45Þ
ð33Þ We now introduce the following expression: b þ 2ÞT =r
ð34Þ where 0 0 0 0 ¼ C 0 b ah ; r 11 ar þ 2C 12 0 0 0 0 bh ¼ C 12 a0h ar þ ðC 22 þ C 023 Þ
ðboundary condition BÞ;
ur ¼ urc þ urp ; ¼/ c þ / p: /
2
0 ðl þ b þ 2Þ 2b 0 rb1 T þ b 0rb T ;r ¼ ½b r h r
or
ð35Þ
If the inner and outer surfaces are traction free, the boundary conditions of inner and outer surfaces can be represented as follows:
H¼
4½cb ðcb l 1Þ þ ca cc þ ðcc þ 1Þðl þ 1Þ2 cc þ 1
ð46Þ
When H is positive, there are k0 = 0, k1 = (l + 1) and two distinct real roots as follows: pffiffiffiffi pffiffiffiffi ðl þ 1Þ þ H ð1 þ 1Þ H ; k3 ¼ ð47Þ k2 ¼ 2 2 When H is negative, there are k0 = 0, k1 = (l + 1) and one pair of conjugate complex roots as follows:
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Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
kJ ¼ aJ jbJ
ð48Þ
For example, when H is positive and l 5 1, urc ðrÞ and c ðrÞ of the homogeneous solution are given by the / following expressions:
ð49Þ
urc ðrÞ ¼
where lþ1 ; aJ ¼ 2
pffiffiffiffiffiffiffiffi H bJ ¼ 2
3 X
F J rkJ ;
J ¼1
Case 1. Real roots for k (k 5 0, l 5 1)
0
Given JN real roots for k except for k0 = 0 if l 5 1, urc ðrÞ c ðrÞ are given by the following expressions: and U urc ðrÞ ¼
JN X
F J rkJ ;
J ¼1
c ðrÞ ¼ U
JN X
M J ðkJ ÞF J rkJ
ð50Þ
J ¼1
3 X
M J ðkJ ÞF J rkJ þ G0
ð56Þ
J ¼1
In order to obtain the particular solution, the series expansions of the Bessel functions are used as shown in Appendix A. Eqs. (16) and (17) can be written as the following expression: T¼
1 X
fa1n ðsÞrnð2mþkÞ þ b1n ðsÞrnð2mþkÞðmþ1Þ g
n¼0
where M J ðkJ Þ ¼
c ðrÞ ¼ C 11 / e01
mþ1 6¼ integer 2mþk 1 X T¼ fa2n ðsÞrnð2mþkÞ þ c2n ðsÞrnð2mþkÞ lnrg if
k2J
þ ðl þ 1ÞkJ ca kJ ðkJ þ l þ 1 cb Þ
ð51Þ
n¼0 c1 X
In Eq. (50), FJ are unknown constants.
þ
Case 2. For k0 = 0 (l 5 1)
n¼0
c ðrÞ are given by the folFor real root k0 ¼ 0; urc ðrÞ and U lowing expressions:
T¼
1 X
fa3n ðsÞrnð2mþkÞðmþ1Þ þ c3n ðsÞrnð2mþkÞðmþ1Þ lnrg
n¼0 c1 X
ð52Þ
mþ1 ¼ positive integer 2mþk ð58Þ
urc ðrÞ ¼ 0; 0 c ðrÞ ¼ e1 G0 U C 011
b2n ðsÞrnð2mþkÞðmþ1Þ if
ð57Þ
þ
b3n ðsÞrnð2mþkÞ if
n¼0 1 X
mþ1 ¼ negative integer 2mþk
fa4n ðsÞrnðkþ3Þ þ c4n ðsÞrnðkþ3Þ lnrg if m ¼ 1
ð59Þ
In Eq. (52), G0 is unknown constant.
T¼
Case 3. For k0 = k1 = 0 (l = 1)
Expressions for the functions ain(s), bin(s) and cin(s) in Eqs. (57)–(60) are omitted here for the sake of brevity. For example, when n(2 m + k) + b + 150 and n(2 m + k) + b m50, the particular solutions urp and p are assumed as the following forms: / 1 X urp ¼ ff1an ðsÞrnð2mþkÞþbþ1 þ f1bn ðsÞrnð2mþkÞmþb g;
Eq. (45) has double root if l = 1. For double root c ðrÞ are given by the following k0 ¼ k1 ¼ 0; urc ðrÞ and U expressions: urc ðrÞ ¼ F 0 ; 0 c ðrÞ ¼ e1 G0 ca F 0 ln r U 0 cb C 11
ð53Þ
1 X
fh1an ðsÞrnð2mþkÞþbþ1 þ h1bn ðsÞrnð2mþkÞmþb g
n¼0
Case 4. Complex roots for k
mþ1 6¼ integer ð61Þ 2mþk 1 X urp ¼ ff2an ðsÞrnð2mþkÞþbþ1 þ f2cn ðsÞrnð2mþkÞþbþ1 ln rg if
c ðrÞ are For one pair of complex roots for k, urc ðrÞ and U given by the following expressions:
n¼0 c1 X
þ ð54Þ
where CJ ¼ Re½M J ðkJ ÞjkJ ¼aJ þjbJ ; XJ ¼ Im½M J ðkJ ÞjkJ ¼aJ þjbJ
n¼0
p ¼ /
In Eq. (53), F0 and G0 are unknown constants.
urc ðrÞ ¼ C 1J raJ cosðbJ ln rÞ þ C 2J raJ sinðbJ ln rÞ; c ðrÞ ¼ C 1J raJ ½CJ cosðbJ ln rÞ XJ sinðbJ ln rÞ U C 2J raJ ½XJ cosðbJ ln rÞ þ CJ sin ðbJ ln rÞ
ð60Þ
n¼0
f2bn ðsÞrnð2mþkÞmþb ;
n¼0
p ¼ /
1 X
fh2an ðsÞrnð2mþkÞþbþ1 þ h2cn ðsÞrnð2mþkÞþbþ1 ln rg
n¼0
ð55Þ
In Eqs. and (55), j, Re[ ] and Im[ ] are imaginary unit pffiffiffiffiffiffi(48) ffi j ¼ 1, real part and imaginary part, respectively. Furthermore, in Eq. (54), C1J and C2J are unknown constants.
þ
c1 X
h2bn ðsÞrnð2mþkÞmþb
n¼0
if
mþ1 ¼ positive integer 2mþk
ð62Þ
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
urp ¼
1 X
ff3an ðsÞrnð2mþkÞmþb þ f3cn ðsÞrnð2mþkÞmþb ln rg
Table 1 Nonhomogeneous parameters
n¼0
þ
c1 X
m
f3bn ðsÞrnð2mþkÞþbþ1 ;
1 X fh3an ðsÞrnð2mþkÞmþb þ h3cn ðsÞrnð2mþkÞmþb ln rg n¼0
þ
c1 X
mþ1 ¼ negative integer if 2mþk 1 X rp ¼ u ff4an ðsÞrnðkþ3Þþbþ1 þ f4cn ðsÞrnðkþ3Þþbþ1 ln rg; p ¼ /
l
0 0 0
0 0 0
0.01 0.01 0.01
Material 2
0 0 0
1 0 1
0 0 0
0.01 0.01 0.01
Material 3
0 0 0
0 0 0
1 0 1
0.01 0.01 0.01
Material 4
0 0 0
0 0 0
0 0 0
1 0.01 1
ð63Þ
n¼0 1 X
b
1 0 1
h3bn ðsÞrnð2mþkÞþbþ1
n¼0
k
Material 1
n¼0
p ¼ /
545
fh4an ðsÞrnðkþ3Þþbþ1 þ h4cn ðsÞrnðkþ3Þþbþ1 ln rg
n¼0
if m ¼ 1
ð64Þ
where fian(s), fibn(s), ficn(s), hian (s), hibn(s) and hicn(s) can be obtained from Eqs. (33) and (34). The particular solutions p for n(2 m + k) + b + 1 = 0 or n(2 m + k) + urp and / b m = 0 are omitted here for the sake of brevity. Then, the stress components and the electric displacement can be evaluated by substituting Eq. (38) into Eqs. (24) and (28), and later into Eqs. (25) and (27). The unknown constants in Eq. (56) are determined so as to satisfy the boundary conditions (36) and (37). 3. Numerical results To illustrate the foregoing analysis, numerical parameters of heat conduction and shape are presented as follows: Ha = Hb = 1.0, T a ¼ 0;
T b ¼ 1;
ra ¼ 0:7
ð65Þ
The hollow sphere is heated from the outer surface. The material constants are taken with reference to cadmium selenide as follows: a0r ¼ 2:458 106 1=K;
k is a nonhomogeneous parameter of the heat capacity per unit volume, b is a nonhomogeneous parameter of the coefficient of linear thermal expansion, and l is a nonhomogeneous parameter of the elastic stiffness constant, the piezoelectric coefficient, and the dielectric constant, respectively. Values given for material 1 correspond to the results for nonhomogeneity of the thermal conductivity; values given for material 2 correspond to the results for nonhomogeneity of the heat capacity per unit volume; values given for material 3 correspond to the results for nonhomogeneity of the coefficient of linear thermal expansion; and values given for material 4 correspond to the results for nonhomogeneity of the elastic stiffness constant, the piezoelectric coefficients, and the dielectric constant. When the nonhomogeneous parameters equal zero, the material constants are constant. As the derived solution breaks down if l = 0, the nonhomogeneous parameter l is taken to be 0.01. In materials 1–4 listed in Table 1, the second values are closely analogous to those for homogeneous material. Figs. 1–10 show the numerical results for the boundary condition A. The results for material 1 are shown in Figs. 1–4. Fig. 1 shows the variation of temperature change in
a0h ¼ 4:396 106 1=K;
C 011 ¼ 83:6 GPa; C 022 ¼ 74:1 GPa; C 012 ¼ 39:3 GPa; C 023 ¼ 45:2 GPa;
e02 ¼ 0:16 C=m2 ;
e01 ¼ 0:347C=m2 ; g01 ¼ 9:03 1011 C2 =Nm2 ;
p01 ¼ 2:94 106 C=m2 K;
kr0 ¼ 12:9 W=mK; c0 ¼ 0:4614 103 J=kg K; q0 ¼ 5:684 103 kg=m3
ð66Þ
The typical values of material properties such as a0, Y0 and d0, used to normalize the numerical data, are a0 ¼ a0h ;
Y 0 ¼ 42:8 GPa;d 0 ¼ 3:92 1012 C=N
ð67Þ
The nonhomogeneous parameters adopted for the numerical calculations are shown in Table 1. In Table 1, m is a nonhomogeneous parameter of the thermal conductivity,
Fig. 1. Variation of temperature change in the radial direction (boundary condition A and material 1).
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Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
Fig. 2. Variation of displacement ur in the radial direction (boundary condition A and material 1).
Fig. 4. Variation of electric potential in the radial direction (boundary condition A and material 1).
the radial direction. As shown in Fig. 1, the temperature rises as the time proceeds and is greatest in a steady state. From Fig. 1, it can be seen that the temperature change on the heated surface increases when the parameter m increases. Fig. 2 shows the variation of displacement ur in the radial direction. It can be seen from Fig. 2 that the displacement ur increases when the parameter m increases. rr and r hh shown in The variations of normal stresses r
Fig. 5. Variation of temperature change in the radial direction (boundary condition A and material 2).
Fig. 3. Variation of thermal stresses in the radial direction (boundary hh . condition A and material 1): (a) normal stress r rr and (b) normal stress r
Fig. 3a and b, respectively. From Fig. 3a, the maximum tensile stress occurs in a steady state inside the hollow sphere without distinction of parameter m. From Fig. 3b, the large compressive stress occurs on the heated surface and the tensile stress occurs on the inner surface without distinction of parameter m. From Fig. 3, the absolute maximum values of the stresses increases when the parameter m increases. Fig. 4 shows the variation of electric potential / in the radial direction. As shown in Fig. 4, the electric potential rises as the time proceeds and is greatest in a steady state. It can be seen that the electric potential increases when the parameter m increases. The results for material 2 are shown in Figs. 5–7. Fig. 5 shows the variation of temperature change in the radial rr and r hh direction. The variations of normal stresses r are shown in Fig. 6a and b, respectively. Fig. 7 shows the in the radial direction. It variation of electric potential / can be seen from Figs. 5–7 that the distributions of the temperature change, thermal stresses and electric potential in a transient state are changed when the parameter k changes, while their distributions in a steady state are same without distinction of parameter k. From Figs. 5–7, the absolute values of the temperature change and electric potential in
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
547
Fig. 6. Variation of thermal stresses in the radial direction (boundary rr and (b) normal stress r hh . condition A and material 2): (a) normal stress r
Fig. 8. Variation of thermal stresses in the radial direction(boundary rr and (b) normal stress r hh . condition A and material 3): (a) normal stress r
Fig. 7. Variation of electric potential in the radial direction (boundary condition A and material 2).
Fig. 9. Variation of electric potential in the radial direction (boundary condition A and material 3).
a transient state increase when the parameter k increases, rr and r hh in a while the absolute values of the stresses r transient state decrease slightly when the parameter k increases. The results for material 3 are shown in Figs. 8 and 9. rr and r hh are shown The variations of normal stresses r in Fig. 8a and b, respectively. Fig. 9 shows the variation in the radial direction. From of electric potential / rr and r hh Fig. 8, the absolute value of the stresses r increases when the parameter b increases. From Fig. 9,
the electric potential increases when the parameter b decreases. The results for material 4 are shown in Fig. 10. The varihh are shown in Fig. 10a rr and r ations of normal stresses r and b, respectively. From Fig. 10, the absolute values of the stresses increase when the parameter l decreases. Fig. 11 shows the numerical result for the boundary condition B. The variation of electric potential in the radial direction for material 1 is shown in Fig. 11. From Fig. 11, it can be seen that the positive maximum electric
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Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549 Table 2 Comparison between the numerical results for boundary condition A and those for boundary condition B in a steady state (material 1) s=1
m
Boundary condition ¼ 0Þ Aðr ¼ ra ; /
Bðr ¼ ra ; Dr ¼ 0Þ
ur ðr ¼ ra Þ
1 0 1
0.5594 0.5069 0.4484
0.5316 0.4817 0.4262
ur ðr ¼ 1Þ
1 0 1
0.6848 0.6206 0.5493
0.6711 0.6083 0.5384
rr ðr ¼ 0:85Þ r
1 0 1
0.03471 0.03454 0.03386
0.03332 0.03328 0.03275
hh ðr ¼ ra Þ r
1 0 1
0.2552 0.2452 0.2314
0.2442 0.2353 0.2226
hh ðr ¼ 1Þ r
1 0 1
0.1600 0.1635 0.1658
0.1539 0.1581 0.1610
Table 3 Comparison between the numerical results for boundary condition A and those for boundary condition B in a transient state (material 1) s = 0.05
m
Boundary condition ¼ 0Þ Aðr ¼ ra ; /
Fig. 10. Variation of thermal stresses in the radial direction (boundary rr and (b) normal stress r hh . condition A and material 4): (a) normal stress r
ur ðr ¼ ra Þ
1 0 1
0.1396 0.1389 0.1375
0.1329 0.1323 0.1310
potential occurs in the inner surface, and the value rises as the time proceeds. The electric potential increases when the parameter m increases as in the case with boundary condition A. In order to examine the influence of the electric boundary condition, the numerical results for boundary condition A and boundary condition B are shown in Tables 2 and 3, which show typical values of the displacement and stresses for material 1. Tables 2 and 3 show the numerical results in
ur ðr ¼ 1Þ
1 0 1
0.1721 0.1712 0.1693
0.1688 0.1679 0.1661
rr ðr ¼ 0:85Þ r
1 0 1
0.03000 0.02739 0.02528
0.02963 0.02705 0.02495
r hh ðr ¼ ra Þ
1 0 1
0.1562 0.1422 0.1312
0.1534 0.1396 0.1286
r hh ðr ¼ 1Þ
1 0 1
0.1768 0.1673 0.1593
0.1754 0.1659 0.1578
Bðr ¼ ra ; Dr ¼ 0Þ
a steady state and a transient state (s = 0.05), respectively. Tables 2 and 3 show that the absolute values of displacement and stresses for boundary condition A are slightly larger than those for boundary condition B. 4. Conclusion
Fig. 11. Variation of electric potential in the radial direction (boundary condition B and material 1).
In the present article, the piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow sphere that has nonhomogeneous thermal and mechanical properties in the radial direction is analyzed. Thermal conductivity, the heat capacity per unit volume, the elastic stiff-
Y. Ootao, Y. Tanigawa / Composite Structures 81 (2007) 540–549
ness constants, the piezoelectric coefficients, the dielectric constant, the coefficient of linear thermal expansion and the pyroelectric constant are expressed as power functions of the radial coordinate. The exact solution for the transient one-dimensional temperature and transient piezothermoelastic response of a functionally graded thermopiezoelectric hollow sphere is obtained. The present solution can serve as a benchmark to the analysis of functionally graded thermopiezoelectric hollow sphere based on various numerical methods. Appendix A. Bessel functions
J c ðxÞ ¼
n 2nþc 1 X ð1Þ ðx=2Þ n!Cðc þ n þ 1Þ n¼0
ðA:1Þ
1 ½cos cp J c ðxÞ J c ðxÞ if c 6¼ integer; sin cp
2 x Y c ðxÞ ¼ J c ðxÞ c þ ln p 2 ! cþn n 1 n X 1 ð1Þ x 2nþc X 1 X 1 þ p n¼0 n!ðc þ nÞ! 2 M M¼1 M M¼1 Y c ðxÞ ¼
c1 1 X ðc n 1Þ! x 2nc p n¼0 n! 2
if c ¼ integer ðA:2Þ
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