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Exact solution of transient thermal stress problem of a multilayered magneto-electro-thermoelastic hollow sphere
Applied Mathematical Modelling 36 (2012) 1431–1443
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Exact solution of transient thermal stress problem of a multilayered magneto-electro-thermoelastic hollow sphere Yoshihiro Ootao ⇑, Masayuki Ishihara Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai 599-8531, Japan
a r t i c l e
i n f o
Article history: Received 23 December 2010 Accepted 31 August 2011 Available online 7 September 2011 Keywords: Thermal stress problem Magneto-electro-thermoelastic Hollow sphere Spherically symmetric problem Transient state
a b s t r a c t This paper presents the theoretical analysis of a multilayered magneto-electro-thermoelastic hollow sphere under unsteady and uniform surface heating. We obtain the exact solution of the transient thermal stress problem of the multilayered magneto-electro-thermoelastic hollow sphere in the spherically symmetric state. As an illustration, we perform numerical calculations of a two-layered composite hollow sphere made of piezoelectric and magnetostrictive materials and investigate the numerical results for temperature change, displacement, stress, and electric and magnetic potential distributions in the transient state are shown in figures. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction It has recently been found that composites made of piezoelectric and magnetostrictive materials exhibit the magnetoelectric effect, which is not seen in piezoelectric or magnetostrictive materials [1–3]. These materials are known as multiferroic composites [4]. These composites exhibit a coupling among magnetic, electric, and elastic fields. It is possible to develop a new system of smart composite materials by combining these piezoelectric and magnetostrictive materials with other structural materials. In the past, various problems in magneto-electro-elastic media that exhibit anisotropic and linear coupling among the magnetic, electric, and elastic fields were analyzed. Examples of static problems are as follows. Pan [5] derived the exact solution of simply supported and multilayered magneto-electro-elastic plates, and Pan and Heyliger [6] derived the exact solution of magneto-electro-elastic laminates in cylindrical bending. Babaei and Chen [7] derived the exact solution of radially polarized and magnetized rotating magneto-electro-elastic hollow and solid cylinders. Ying and Wang [8] derived the exact solution of rotating magneto-electro-elastic composite hollow cylinders. Wang et al. [9] derived an analytical solution of a multilayered magneto-electro-elastic circular plate under simply supported boundary conditions. Examples of dynamic problems are as follows. Wang and Ding [10] analyzed the transient responses of a magneto-electro-elastic hollow sphere and a magneto-electro-elastic composite hollow sphere [11] subjected to spherically symmetric dynamic loads. Anandkumar et al. [12] analyzed the free vibration behavior of multiphase and layered magneto-electro-elastic beams. Examples of thermal stress problems are as follows. Sunar et al. [13] analyzed thermopiezomagnetic smart structures and Kumaravel et al. [14] analyzed a three-layered electro-magneto-elastic strip under steady state conditions using the finite element method. Hou et al. [15] obtained 2D fundamental solutions of a steady point heat source in infinite and semi-infinite orthotropic electro-magneto-thermo-elastic planes and obtained Green’s function for a steady point heat source on the
⇑ Corresponding author. Tel.: +81 72 254 9208; fax: +81 72 254 9904. E-mail address: [email protected] (Y. Ootao). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.08.043
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surface of a semi-infinite transversely isotropic electro-magneto-thermo-elastic material [16]. Xiong and Ni [17] obtained 2D Green’s functions for semi-infinite transversely isotropic electro-magneto-thermo-elastic composites. Gao et al. [18] analyzed the problem of collinear cracks in an electro-magneto-thermo-elastic solid subjected to uniform heat flow at infinity. These studies, however, treated thermal stress problems only under steady temperature distribution. It is well known that thermal stress distributions in a transient state show significant and large response values as compared to those in a steady state. Therefore, transient thermoelastic problems are important. With regard to transient thermal stress problems, Wang and Niraula [19] analyzed transient thermal fracture in transversely isotropic electro-magneto-elastic cylinders. The exact solution of a transient analysis of multilayered magneto-electro-thermoelastic strip subjected to nonuniform heat supply was also obtained [20]. However, to the best of the authors’ knowledge, the exact analysis of a multilayered magnetoelectro-thermoelastic hollow sphere under unsteady heat supply has not yet been reported. Here, we present the derivation of an exact solution of the transient thermal stress problem of a multilayered composite hollow sphere made of magnetoelectro-thermoelastic materials under uniform surface heating in a spherically symmetric state. We assumed that the magneto-electro-thermoelastic materials are polarized and magnetized in the radial direction. 2. Analysis We considered a multilayered composite hollow sphere made of spherical isotropic and linear magneto-electrothermoelastic materials. The hollow sphere’s inner and outer radii are denoted by a and b, respectively. ri is the outer radius of the ith layer. Throughout this article, indices i (=1, 2, . . . , N) are associated with the ith layer from the inner side of a composite hollow sphere. 2.1. Heat conduction problem We assumed that the multilayered hollow sphere is initially at zero temperature and its the inner and outer surfaces are suddenly heated by surrounding media having constant temperatures Ta and Tb with relative heat transfer coefficients ha and hb, respectively. Then, the temperature distribution is one-dimensional and the transient heat conduction equation for the ith layer is written in the following form:
! @T i @ 2 T i 2 @T i ri ; ¼j þ r @r @s @r 2
i ¼ 1; . . . ; N:
ð1Þ
The initial and thermal boundary conditions in dimensionless form are
s ¼ 0; T i ¼ 0; i ¼ 1; 2; . . . ; N;
ð2Þ
@T 1 Ha T 1 ¼ Ha T a ; @r r ¼ Ri ; T i ¼ T iþ1 ; i ¼ 1; 2; . . . ; N 1; ; r ¼ a
r ¼ Ri ; r ¼ 1;
kri @T i ¼ kr;iþ1 @T iþ1 ; @r @r @T N þ Hb T N ¼ Hb T b : @r
ð3Þ ð4Þ
i ¼ 1; 2; . . . ; N 1;
ð5Þ ð6Þ
In Eqs. (1)–(6), we introduced the following dimensionless values:
Þ ¼ ðr; r i ; aÞ=b; ðT i ; T a ; T b Þ ¼ ðT i ; T a ; T b Þ=T 0 ; ðr ; Ri ; a jri ¼ jri =j0 ; kri ¼ kri =k0 ; ðHa ; Hb Þ ¼ ðha ; hb Þb;
s ¼ j0 t=b2 ;
ð7Þ
where Ti is the temperature change; t is time; and T0, k0 and j0 are typical values of temperature, thermal conductivity, and thermal diffusivity, respectively. Introducing the Laplace transform with respect to the variable s, the solution of Eq. (1) can be obtained so as to satisfy the conditions (2)–(6). This solution is written as follows:
B0 1 Ti ¼ A0i þ i r F
! þ
X1 2 exp l2j s h j¼1
0
lj D ðlj Þ
i i y ðb l r Þ ; Ai j0 ðbi ljr Þ þ B i j 0
i ¼ 1; 2; . . . ; N;
ð8Þ
where j0() and y0() are zeroth-order Spherical Bessel functions of the first and second kind, respectively. Furthermore, D and F are the determinants of 2N 2N matrice [akl] and [ekl], respectively; the coefficients Ai and Bi are defined as determinants of a matrix similar to the coefficient matrix [akl], in which the (2i 1)th column or 2ith column is replaced with the constant vector {ck}, respectively. Similarly, the coefficients A0i and B0i are defined as determinants of a matrix similar to the coefficient matrix [ekl], in which the (2i 1)th column or 2ith column is replaced with the constant vector {ck}, respectively. The nonzero elements of the coefficient matrices [akl] and [ekl] and the constant vector {ck} are given as
Y. Ootao, M. Ishihara / Applied Mathematical Modelling 36 (2012) 1431–1443
Þ þ Ha j0 ðb1 la Þ; a1;1 ¼ b1 lj1 ðb1 la
Þ þ Ha y0 ðb1 la Þ; a1;2 ¼ b1 ly1 ðb1 la
a2N;2N1 ¼ Hb j0 ðbN lÞ bN lj1 ðbN lÞ; a2i;2i1 ¼ j0 ðbi lRi Þ;
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a2N;2N1 ¼ Hb y0 ðbN lÞ bN ly1 ðbN lÞ;
a2i;2i ¼ y0 ðbi lRi Þ;
a2i;2iþ1 ¼ j0 ðbiþ1 lRi Þ;
a2i;2iþ2 ¼ y0 ðbiþ1 lRi Þ;
ð9Þ
a2iþ1;2i1 ¼ kri bi lj1 ðbi lRi Þ; a2iþ1;2i ¼ kri bi ly1 ðbi lRi Þ;
a2iþ1;2iþ1 ¼ kr;iþ1 biþ1 lj1 ðbiþ1 lRi Þ;
a2iþ1;2iþ2 ¼ kr;iþ1 biþ1 ly1 ðbiþ1 lRi Þ; i ¼ 1; 2; . . . ; N 1 1 Þ; e2N;2N1 ¼ Hb ; e2N;2N ¼ Hb 1 e1;1 ¼ Ha ; e1;2 ¼ 2 ð1 þ Ha a a e2i;2i1 ¼ 1; e2iþ1;2i ¼
kri
e2i;2i ¼
; 2
Ri
c1 ¼ Ha T a ;
1 ; Ri
e2i;2iþ1 ¼ 1;
e2iþ1;2iþ1 ¼
kr;iþ1 R2i
e2i;2iþ2 ¼
1 ; Ri
ð10Þ ð11Þ
ð12Þ
i ¼ 1; 2; . . . ; N 1
;
c2N ¼ Hb T b ;
ð13Þ
In Eq. (8), D0 (lj) is
D0 ð l j Þ ¼
dD dll¼lj
ð14Þ
and lj is the jth positive root of the following transcendental equation:
DðlÞ ¼ 0:
ð15Þ
2.2. Thermoelastic problem We developed the analysis of a multilayered magneto-electro-thermoelastic hollow sphere as a spherically symmetric state. The displacement–strain relations are expressed in dimensionless form as follows:
erri ¼ u ri ;r ;
ehhi ¼ e//i ¼ u ri =r ;
crhi ¼ cr/i ¼ ch/i ¼ 0;
ð16Þ
where the comma denotes partial differentiation with respect to the variable that follows. For the spherical isotropic and linear magneto-electro-thermoelastic materials, the constitutive relations are expressed in dimensionless form as follows:
r rri ¼ C 11i erri þ 2C 12i ehhi bri T i e1i Eri q1i Hri ; r hhi ¼ r //i ¼ C 12i erri þ ðC 22i þ C 23i Þehhi bhi T i e2i Eri q2i Hri ;
ð17Þ
ri ¼ C 11i a ri þ 2C 12i a hi ; b ri þ ðC 22i þ C 23i Þa hi : bhi ¼ C 12i a
ð18Þ
where
The constitutive equations for the electric and the magnetic fields in dimensionless form are given as
H þp 1i Eri þ d 1i T i ; Dri ¼ e1i erri þ 2e2i ehhi þ g 1i ri e þ 2q e þ d E þ l H þm T: B ¼q ri
1i rri
2i hhi
1i ri
1i
ri
1i i
ð19Þ ð20Þ
The relation between the electric field intensity and the electric potential /i in dimensionless form is defined as
i ;r : Eri ¼ /
ð21Þ
The relation between the magnetic field intensity and the magnetic potential wi in dimensionless form is defined as s
i ;r : Hri ¼ w
ð22Þ
The equilibrium equation in the radial direction is expressed in dimensionless form as follows:
r rri ;r þ 2ðr rri r hhi Þ=r ¼ 0:
ð23Þ
If the electric charge density is zero, the equations of electrostatics and magnetostatics are expressed in dimensionless form as follows:
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Dri ;r þ 2Dri =r ¼ 0;
ð24Þ
Bri ;r þ 2Bri =r ¼ 0:
ð25Þ
In Eqs. (16)–(25), the following dimensionless values are introduced:
rkli ðe ; c Þ uir a C kli Þ ¼ kli kli ; u ki ¼ ki ; C kli ¼ kli ; ri ¼ ; ðekli ; c ; a a0 Y 0 T 0 a0 T 0 a0 T 0 b a0 Y0
r kli ¼ Dri ¼
Dri
a0 Y 0 T 0 jd0 j
Bri ¼
;
eki ; Y 0 jd0 j
g 1i ¼
¼ j0 d1i ; d 1i b
1i ¼ p
eki ¼
Hri ¼
Bri jd0 jj0 ; ba0 T 0
g1i Y 0 jd0 j
2
ki ¼ q
;
p1i
a0 Y 0 jd0 j
;
i ¼ /i jd0 j ; / a0 T 0 b qki j0 jd0 j ; b
1i ¼ m
m1i j0 jd0 j ; ba0
i ¼ w
l 1i ¼
wi ; jd0 jj0 a0 Y 0 T 0
l1i j20 jd0 j2 Y 0
Eri ¼
2
b
;
ð26Þ
Eri jd0 j ; a0 T 0
Hri b ; jd0 jj0 a0 Y 0 T 0
where rkli are the stress components; (ekli, ckli) are the strain components; uri is the displacement in the r direction; aki are the coefficients of linear thermal expansion; Ckli are the elastic stiffness constants; Dri is the electric displacement in the r direction; Bri is the magnetic flux density in the r direction; eki are the piezoelectric coefficients; g1i is the dielectric constant; p1i is the pyroelectric constant; qki are the piezomagnetic coefficients; l1i is the magnetic permeability coefficient; d1i is the magnetoelectric coefficient; m1i is the pyromagnetic constant; and a0, Y0, and d0 are typical values of the coefficient of linear thermal expansion, Young’s modulus and piezoelectric modulus, respectively. Substituting Eqs. (16), (21) and (22) into Eqs. (17), (19) and (20), and later into Eqs. (23)–(25), the governing equations of the displacement uri, electric potential /i, and magnetic potential wi in the dimensionless form are written as
i ;r^r þ 2ðe1i e2i Þ/ i ;rr þ 2ðq i ;r r 1 þ q i ;r r 1 1i w 1i q 2i Þw rir 2 þ e1i / ri ;rr þ 2C 11i u ri ;r r1 þ 2ðC 12i C 22i C 23i Þu C 11i u ri b hi ÞT ir1 þ b ri T i ;r ; ¼ 2ðb
ð27Þ
w r1 ¼ p i ;rr 2g i ;r r 1 d e1i u ri ;rr þ 2ðe1i þ e2i Þu rir 2 g ri ;r r 1 þ 2e2i u 1i ðT i ;r þ 2T ir1 Þ; 1i / 1i / 1i i ;r^r 2d1i wi ;r
ð28Þ
/ r1 l i ;rr 2l i ;r r 1 ¼ m 1i u 1i þ q 2i Þu 2i u ri ;r r 1 þ 2q 1i w 1i w 1i ðT i ;r þ 2T ir 1 Þ: ri ;rr þ 2ðq rir2 d q 1i i ;rr 2d1i /i ;r
ð29Þ
If the inner and outer surfaces of the multilayered magneto-electro-thermoelastic hollow sphere are traction free, and the interfaces of each adjoining layer are perfectly bonded, then the boundary conditions of inner and outer surfaces and the conditions of continuity at the interfaces can be represented as follows:
; r rr1 ¼ 0; r ¼ a r ¼ Ri ; r rri ¼ r rr;iþ1 ; r ¼ 1;
ri ¼ u r;iþ1 ; u
i ¼ 1; . . . ; N 1;
ð30Þ
r rrN ¼ 0:
The boundary conditions in the radial direction for the electric and magnetic fields are expressed as
1 ¼ 0; w 1 ¼ 0; Dr1 ¼ 0; Br1 ¼ 0 or / i ¼ / iþ1 ; w i ¼ w iþ1 ; r ¼ Ri ; Dri ¼ Dr;iþ1 ; Bri ¼ Br;iþ1 ; / r ¼ 1; DrN ¼ 0; BrN ¼ 0 or /N ¼ 0; wN ¼ 0:
; r ¼ a
i ¼ 1; . . . ; N 1;
ð31Þ
The solutions of Eqs. (27)–(29) are assumed in the following form:
rci þ u rpi ; ri ¼ u u
i ¼ / ci þ / pi ; /
i ¼ w ci þ w pi w
ð32Þ
In Eq. (32), the first term on the right-hand side gives the homogeneous solution and the second term gives the particular solution. The homogeneous solutions of Eq. (32) can be expressed as follows:
rci ¼ C 1ir 1 þ C 2irki2 þ C 3ir ki3 ; u ci ¼ C 11i ðC 6i þ C 7ir1 þ g C 2ir ki2 þ g C 3ir ki3 Þ; / 1i 2i e1i ci ¼ C 11i ðC 4i þ C 5ir1 þ g C 2ir ki2 þ g C 3ir ki3 Þ; w 3i 4i 1i q
ð33Þ
Y. Ootao, M. Ishihara / Applied Mathematical Modelling 36 (2012) 1431–1443
1435
where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4b1i =b3i ; 2 # bqi ci bei bdi ð c b b c þ b b Þ ; b1i ¼ ai ci þ b2ei bei di qi i ei di i di ci b2di ki2 ; ki3 ¼ "
1
b3i ¼ 1 þ ci þ
ai ¼
ðci bdi Þ2 di ci b2di
;
2ðC 22i þ C 23i C 12i Þ C 11i
C g ci ¼ 11i2 1i ; e1i g 1i ¼ g 2i ¼
;
bei ¼
2e2i ; e1i
bqi ¼
2i 2q ; 1i q
bdi ¼
1i C 11i d ; 1i e1i q
1i C 11i l ; 21i q 2 ki2 þ ki2 þ bei ðki2 þ 1Þ bdi ki2 a3i ;
di ¼
1
ci ki2 ðki2 þ 1Þ 1
ci ki3 ðki3 þ 1Þ
ð34Þ
2 ki3 þ ki3 þ bei ðki3 þ 1Þ bdi ki3 a4i ;
g 3i ¼ a3i =ðki2 þ 1Þ; g 4i ¼ a4i =ðki3 þ 1Þ; ki2 þ 1 a3i ¼ ½ðci bdi Þki2 þ bqi ci bei bdi ; ki2 di ci b2di ki3 þ 1 a4i ¼ ½ðci bdi Þki3 þ bqi ci bei bdi ki3 di ci b2di In Eq. (33), Cki(k = 1, 2, . . . , 7) are unknown constants. We have the following relation:
ai C 1i þ bei C 7i þ bqi C 5i ¼ 0:
ð35Þ
In order to obtain the particular solutions, series expansions of Sphere Bessel functions given in Eq. (8) are used. Eq. (8) can be written in the following way:
T i ðr i ; sÞ ¼
1 X
ain ðsÞr 2n þ bin ðsÞr 2n1 ;
ð36Þ
n¼0
where
2n 1 2 exp l2j s ð1Þn bi lj X A0i ain ðsÞ ¼ d0n þ Ai ; F ð2n þ 1Þ! lj D0 ðlj Þ j¼1 1 2 exp l2j s ð1Þnþ1 ðbi lj Þ2n1 X B0i Bi : bin ðsÞ ¼ d0n þ F ð2nÞ! lj D0 ðlj Þ j¼1
Fig. 1. Variation of temperature change in the radial direction (Case 1, R1 = 0.85).
ð37Þ
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r in the radial direction (Case 1, R1 = 0.85). Fig. 2. Variation of displacement u
rr and (b) normal stress r hh . Fig. 3. Variation of thermal stresses in the radial direction (Case 1, R1 = 0.85): (a) normal stress r
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Fig. 4. Variation of electric potential in the radial direction (Case 1, R1 = 0.85).
Fig. 5. Variation of magnetic potential in the radial direction (Case 1, R1 = 0.85).
pi , and w pi are obtained in the following forms: rpi ; / Here, d0n is the Kronecker delta. The particular solutions u
rip ¼ u
1 X
fani ðsÞr2nþ1 þ fbni ðsÞr 2n ;
n¼0
ip ¼ / ip ¼ w
1 X n¼0 1 X
hani ðsÞr 2nþ1 þ hbni ðsÞr2n þ hc0i ðsÞ ln r ;
ð38Þ
g ani ðsÞr 2nþ1 þ g bni ðsÞr2n þ g c0i ðsÞ ln r :
n¼0
Expressions for fani(s), fbni(s), hani(s), hbni(s), hc0i(s), gani(s), gbni(s), and gc0i(s) in Eq. (38) have been omitted here for brevity. Then, the stress components, electric displacement, and the magnetic flux density can be evaluated from Eq. (33). Details of the solutions are omitted from here for brevity. The unknown constants in the homogeneous solutions are determined so as to satisfy the boundary conditions in (30) and (31). 3. Numerical results In order to illustrate the foregoing analysis, we consider a two-layered hollow sphere composed of piezoelectric and magnetostrictive layers. The piezoelectric layer is made up of BaTiO3, and the magnetostrictive layer is made up of CoFe2O4. Two
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Fig. 6. Variation of temperature change in the radial direction (Case 2, R1 = 0.85).
r in the radial direction (Case 2, R1 = 0.85). Fig. 7. Variation of displacement u
kinds of two-layered hollow sphere are investigated. Case 1 shows the stacking sequence BaTiO3/CoFe2O4 and Case 2 shows the stacking sequence CoFe2O4/BaTiO3. We assume that the outer surface of the two-layered hollow sphere is heated. Then, numerically calculable parameters of the heat condition and shape are presented as follows:
Ha ¼ Hb ¼ 1:0; T a ¼ 0; T b ¼ 1; N ¼ 2; ¼ 0:7; R1 ¼ 0:75; 0:8; 0:85; 0:9; 0:95; b ¼ 0:01m a
ð39Þ
The following are material constants considered for BaTiO3 [20]:
ah ¼ a/ ¼ 15:7 106 1=K; ar ¼ 6:4 106 1=K; C 22 ¼ 166 GPa; C 23 ¼ 77 GPa; C 12 ¼ 78 GPa; C 11 ¼ 162 GPa e2 ¼ 4:4 C=m2 ;
e1 ¼ 18:6 C=m2 ;
p1 ¼ 2 104 C2 =m2 K;
jr ¼ 0:88 106 m2 =s:
g1 ¼ 12:6 109 C2 =Nm2 ;
l1 ¼ 10 106 Ns2 =C2 ; kr ¼ 2:5 W=mK;
ð40Þ
Y. Ootao, M. Ishihara / Applied Mathematical Modelling 36 (2012) 1431–1443
rr and (b) normal stress r hh . Fig. 8. Variation of thermal stresses in the radial direction (Case 2, R1 = 0.85): (a) normal stress r
Fig. 9. Variation of electric potential in the radial direction (Case 2, R1 = 0.85).
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Fig. 10. Variation of magnetic potential in the radial direction (Case 2, R1 = 0.85).
rr and (b) normal stress r hh . Fig. 11. Variation of thermal stresses in the radial direction (Case 1, s = 1): (a) normal stress r
Y. Ootao, M. Ishihara / Applied Mathematical Modelling 36 (2012) 1431–1443
1441
Fig. 12. Variation of electric potential in the radial direction (Case 1, s = 1).
Fig. 13. Variation of magnetic potential in the radial direction (Case 1, s = 1).
The corresponding constants for CoFe2O4 are
ar ¼ ah ¼ a/ ¼ 10 106 1=K; C 22 ¼ 286 GPa; C 23 ¼ 173 GPa; C 12 ¼ 170:5 GPa; C 11 ¼ 269:5 GPa; q2 ¼ 580:3 N=Am; q1 ¼ 699:7 N=Am; g1 ¼ 0:093 109 C2 =Nm2 ; 4
p1 ¼ 2 10
2
2
6
2
C =m K;
jr ¼ 0:77 10
l1 ¼ 157 10
6
2
2
Ns =C ;
ð41Þ
kr ¼ 3:2 W=mK;
m =s:
The typical values of material parameters such as j0, k0, a0, Y0, and d0, used to normalize the numerical data, based on those of BaTiO3 are as follows:
j0 ¼ jr ; k0 ¼ kr ; a0 ¼ ar ; Y 0 ¼ 116 GPa; d0 ¼ 78 1012 C=N:
ð42Þ
In the numerical calculations, the boundary conditions at the surfaces for the electric and magnetic fields are expressed as
; r ¼ a r ¼ 1;
Dr1 ¼ 0; Br1 ¼ 0; N ¼ 0: N ¼ 0; w /
ð43Þ
The numerical results for Case 1 and R1 = 0.85 are shown in Figs. 1–5. Fig. 1 shows the variation of temperature change along r along the radial direction. From Figs. 1 and 2, it is clear that the radial direction. Fig. 2 shows the variation of displacement u the temperature and displacement increase with time and have the largest values in steady state. Fig. 3(a) and (b) shows
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rr and r hh , respectively, along the radial direction. Fig. 3(a) reveals that the maximum tensile variations of thermal stresses r stress occurs in the transient state and the maximum compressive stress occurs in the steady state. From Fig. 3(b), it is clear that the compressive stress occurs in the first layer and tensile stress occurs in the second layer. Figs. 4 and 5 show variations and magnetic potential w, respectively, along the radial direction. Fig. 4 reveals that the absolute value of electric potential / of the electric potential increases with time, except during the early stage of heating, and attains its maximum value in the steady state. The electric potential is almost zero in the second layer, i.e. the magnetostrictive layer. From Fig. 5, it is clear that the absolute value of the magnetic potential increases with time and attains its maximum value in the steady state. The magnetic potential is almost constant in the first layer, i.e. the piezoelectric layer. The numerical results for Case 2 and R1 = 0.85 are shown in Figs. 6–10. Fig. 6 shows the variation of temperature change r along the radial direction. From Figs. 1, 2, 6 and 7, it is along the radial direction. Fig. 7 shows the variation of displacement u r of Case 1 are larger than those of Case 2. Fig. 8(a) and (b) shows the clear that the temperature increase and displacement u rr and r hh , respectively, along the radial direction. From Fig. 8(a), it is clear that the maximum variations of thermal stresses r tensile stress occurs in the steady state. Fig. 8(b) reveals that tensile stress occurs in the first layer and compressive stress and magnetic potential w, respectively, occurs in the second layer. Figs. 9 and 10 show the variations of electric potential / along the radial direction. From these figures, it is clear that the absolute values of the electric and the magnetic potential increases with time and attain their maximum value in the steady state. The electric potential is almost constant in the first layer, i.e. the magnetostrictive layer. In contrast, the magnetic potential is almost zero in the second layer, i.e. the piezoelectric layer. In order to assess the influence of the position of the interface between both layers, numerical results for Case 1 and R1 = 0.75, 0.8, 0.85, 0.9, 0.95 were obtained; these results are shown in Figs. 11–13. Fig. 11(a) and (b) shows the variations rr and r hh , respectively, in the steady state. Figs. 12 and 13 show the variations of electric and magnetic of thermal stresses r rr potential, respectively, in the steady state. From Fig. 11(a) and (b), it is clear that the distribution of the thermal stress r hh increases with an increase changes substantially with a change in the parameter R1 whereas the maximum tensile stress r in R1. It can be seen from Figs. 12 and 13 that the absolute values of electric potential increase and those of magnetic potential decrease with an increase in R1. 4. Conclusion In this study, we obtained the exact solution of the transient thermal stress problem of a multilayered magnetoelectro-thermoelastic hollow sphere under uniform surface heating as a spherically symmetric state. As an illustration, we carried out numerical calculations for a two-layered hollow sphere composed of piezoelectric and magnetostrictive materials and examined its behavior in the transient state in terms of temperature change, displacement, stress, and electric and magnetic potential distributions. Furthermore, the effects of the stacking sequence and position of the interface were investigated. Though numerical calculation were carried out for a two-layered hollow sphere, numerical calculation for multilayered magneto-electro-thermoelastic hollow sphere with an arbitrary number of layer and arbitrary stacking sequence can be carried out. The present solution can serve as a benchmark to the analysis of magneto-electro-thermoelastic hollow sphere based on various numerical methods. References [1] G. Harrshe, J.P. Dougherty, R.E. Newnhan, Theoretical modeling of multilayer magnetoelectric composite, Int. J. Appl. Electromagn. Mater. 4 (1993) 145–159. [2] C.W. Nan, Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases, Phys. Rev. B50 (1994) 6082–6088. [3] Y. Benveniste, Magnetoelectric effect in fibrous composites with piezoelecric and piezomagnetic, Phases. Phys. Rev. B 51 (1995) 16424– 16427. [4] C.W. Nan, M.I. Bichurin, S. Dong, D. Viehland, G. Srinivasan, Multiferroic magnetoelectric composites: historical perspective, status, and future directions, J. Appl. Phys. 103 (2008) 031101. [5] E. Pan, Exact solution for simply supported and multilayered magneto-electro-elastic plates, Trans. ASME J. Appl. Mech. 68 (2001) 608–618. [6] E. Pan, P.R. Heyliger, Exact solutions for magneto-electro-elastic laminates in cylindrical bending, Int. J. Solids Struct. 40 (2003) 6859–6876. [7] M.H. Babaei, Z.T. Chen, Exact solutions for radially polarized and magnetized magnetoelectroelastic rotating cylinders, Smart Mater. Struct. 17 (2008) 025035. [8] J. Ying, H.M. Wang, Magnetoelectroelastic fields in rotating multiferroic composite cylindrical structures, J. Zhejiang Univ. Sci. A 10 (2009) 319–326. [9] R. Wang, Q. Han, E. Pan, An analytical solution for a multilayered magneto-electro-elastic circular plate under simply supported lateral boundary conditions, Smart Mater. Struct. 19 (2010) 065025. [10] H.M. Wang, H.J. Ding, Transient responses of a magneto-electro-elastic hollow sphere for fully coupled spherically symmetric problem, Eur. J. Mech. A/ Solids 25 (2006) 965–980. [11] H.M. Wang, H.J. Ding, Radial vibration of piezoelectric/magnetostrictive composite hollow sphere, J. Sound Vib. 307 (2007) 330–348. [12] R. Anandkumar, R. Annigeri, N. Ganesan, S. Swarnamani, Free vibration behavior of multiphase and layered magneto-electro-elastic beam, J. Sound Vib. 299 (2007) 44–63. [13] M. Sunar, A.Z. Al-Garni, M.H. Ali, R. Kahraman, Finite element modeling of thermopiezomagnetic smart structures, AIAA J. 40 (2001) 1846–1851. [14] A. Kumaravel, N. Ganesan, R. Sethurraman, Steady-state analysis of a three-layered electro-magneto-elastic strip in a thermal environment, Smart. Mater. Struct. 16 (2007) 282–295. [15] P.F. Hou, T. Yi, L. Wang, 2D general solution and fundamental solution for orthotropic electro-magneto-elastic materials, J. Therm. Stresses 31 (2008) 807–822.
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[16] P.F. Hou, A.Y. Leung, H.J. Ding, A point heat source on the surface of a semi-infinite transversely isotropic electro-magneto-thermo-elastic material, Int. J. Eng. Sci. 46 (2008) 273–285. [17] S.M. Xiong, G.Z. Ni, 2D green’s functions for semi-infinite transversely isotropic electro-magneto-thermo-elastic composite, J. Magnetism Magn. Mater. 321 (2009) 1867–1874. [18] C.F. Gao, H. Kessler, H. Balke, Fracture analysis of electromagnetic thermoelastic solids, Eur. J. Mech. A/Solids 22 (2003) 433–442. [19] B.L. Wang, O.P. Niraula, Transient thermal fracture analysis of transversely isotropic magneto-electro-elastic materials, J. Therm. Stresses 30 (2007) 297–317. [20] Y. Ootao, Y. Tanigawa, Transient analysis of multilayered magneto-electro-thermoelastic strip due to nonuniform heat supply, Comp. Struct. 68 (2005) 471–480.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Exact solution of transient thermal stress problem of a multilayered magneto-electro-thermoelastic hollow sphere Yoshihiro Ootao ⇑, Masayuki Ishihara Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai 599-8531, Japan
a r t i c l e
i n f o
Article history: Received 23 December 2010 Accepted 31 August 2011 Available online 7 September 2011 Keywords: Thermal stress problem Magneto-electro-thermoelastic Hollow sphere Spherically symmetric problem Transient state
a b s t r a c t This paper presents the theoretical analysis of a multilayered magneto-electro-thermoelastic hollow sphere under unsteady and uniform surface heating. We obtain the exact solution of the transient thermal stress problem of the multilayered magneto-electro-thermoelastic hollow sphere in the spherically symmetric state. As an illustration, we perform numerical calculations of a two-layered composite hollow sphere made of piezoelectric and magnetostrictive materials and investigate the numerical results for temperature change, displacement, stress, and electric and magnetic potential distributions in the transient state are shown in figures. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction It has recently been found that composites made of piezoelectric and magnetostrictive materials exhibit the magnetoelectric effect, which is not seen in piezoelectric or magnetostrictive materials [1–3]. These materials are known as multiferroic composites [4]. These composites exhibit a coupling among magnetic, electric, and elastic fields. It is possible to develop a new system of smart composite materials by combining these piezoelectric and magnetostrictive materials with other structural materials. In the past, various problems in magneto-electro-elastic media that exhibit anisotropic and linear coupling among the magnetic, electric, and elastic fields were analyzed. Examples of static problems are as follows. Pan [5] derived the exact solution of simply supported and multilayered magneto-electro-elastic plates, and Pan and Heyliger [6] derived the exact solution of magneto-electro-elastic laminates in cylindrical bending. Babaei and Chen [7] derived the exact solution of radially polarized and magnetized rotating magneto-electro-elastic hollow and solid cylinders. Ying and Wang [8] derived the exact solution of rotating magneto-electro-elastic composite hollow cylinders. Wang et al. [9] derived an analytical solution of a multilayered magneto-electro-elastic circular plate under simply supported boundary conditions. Examples of dynamic problems are as follows. Wang and Ding [10] analyzed the transient responses of a magneto-electro-elastic hollow sphere and a magneto-electro-elastic composite hollow sphere [11] subjected to spherically symmetric dynamic loads. Anandkumar et al. [12] analyzed the free vibration behavior of multiphase and layered magneto-electro-elastic beams. Examples of thermal stress problems are as follows. Sunar et al. [13] analyzed thermopiezomagnetic smart structures and Kumaravel et al. [14] analyzed a three-layered electro-magneto-elastic strip under steady state conditions using the finite element method. Hou et al. [15] obtained 2D fundamental solutions of a steady point heat source in infinite and semi-infinite orthotropic electro-magneto-thermo-elastic planes and obtained Green’s function for a steady point heat source on the
⇑ Corresponding author. Tel.: +81 72 254 9208; fax: +81 72 254 9904. E-mail address: [email protected] (Y. Ootao). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.08.043
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surface of a semi-infinite transversely isotropic electro-magneto-thermo-elastic material [16]. Xiong and Ni [17] obtained 2D Green’s functions for semi-infinite transversely isotropic electro-magneto-thermo-elastic composites. Gao et al. [18] analyzed the problem of collinear cracks in an electro-magneto-thermo-elastic solid subjected to uniform heat flow at infinity. These studies, however, treated thermal stress problems only under steady temperature distribution. It is well known that thermal stress distributions in a transient state show significant and large response values as compared to those in a steady state. Therefore, transient thermoelastic problems are important. With regard to transient thermal stress problems, Wang and Niraula [19] analyzed transient thermal fracture in transversely isotropic electro-magneto-elastic cylinders. The exact solution of a transient analysis of multilayered magneto-electro-thermoelastic strip subjected to nonuniform heat supply was also obtained [20]. However, to the best of the authors’ knowledge, the exact analysis of a multilayered magnetoelectro-thermoelastic hollow sphere under unsteady heat supply has not yet been reported. Here, we present the derivation of an exact solution of the transient thermal stress problem of a multilayered composite hollow sphere made of magnetoelectro-thermoelastic materials under uniform surface heating in a spherically symmetric state. We assumed that the magneto-electro-thermoelastic materials are polarized and magnetized in the radial direction. 2. Analysis We considered a multilayered composite hollow sphere made of spherical isotropic and linear magneto-electrothermoelastic materials. The hollow sphere’s inner and outer radii are denoted by a and b, respectively. ri is the outer radius of the ith layer. Throughout this article, indices i (=1, 2, . . . , N) are associated with the ith layer from the inner side of a composite hollow sphere. 2.1. Heat conduction problem We assumed that the multilayered hollow sphere is initially at zero temperature and its the inner and outer surfaces are suddenly heated by surrounding media having constant temperatures Ta and Tb with relative heat transfer coefficients ha and hb, respectively. Then, the temperature distribution is one-dimensional and the transient heat conduction equation for the ith layer is written in the following form:
! @T i @ 2 T i 2 @T i ri ; ¼j þ r @r @s @r 2
i ¼ 1; . . . ; N:
ð1Þ
The initial and thermal boundary conditions in dimensionless form are
s ¼ 0; T i ¼ 0; i ¼ 1; 2; . . . ; N;
ð2Þ
@T 1 Ha T 1 ¼ Ha T a ; @r r ¼ Ri ; T i ¼ T iþ1 ; i ¼ 1; 2; . . . ; N 1; ; r ¼ a
r ¼ Ri ; r ¼ 1;
kri @T i ¼ kr;iþ1 @T iþ1 ; @r @r @T N þ Hb T N ¼ Hb T b : @r
ð3Þ ð4Þ
i ¼ 1; 2; . . . ; N 1;
ð5Þ ð6Þ
In Eqs. (1)–(6), we introduced the following dimensionless values:
Þ ¼ ðr; r i ; aÞ=b; ðT i ; T a ; T b Þ ¼ ðT i ; T a ; T b Þ=T 0 ; ðr ; Ri ; a jri ¼ jri =j0 ; kri ¼ kri =k0 ; ðHa ; Hb Þ ¼ ðha ; hb Þb;
s ¼ j0 t=b2 ;
ð7Þ
where Ti is the temperature change; t is time; and T0, k0 and j0 are typical values of temperature, thermal conductivity, and thermal diffusivity, respectively. Introducing the Laplace transform with respect to the variable s, the solution of Eq. (1) can be obtained so as to satisfy the conditions (2)–(6). This solution is written as follows:
B0 1 Ti ¼ A0i þ i r F
! þ
X1 2 exp l2j s h j¼1
0
lj D ðlj Þ
i i y ðb l r Þ ; Ai j0 ðbi ljr Þ þ B i j 0
i ¼ 1; 2; . . . ; N;
ð8Þ
where j0() and y0() are zeroth-order Spherical Bessel functions of the first and second kind, respectively. Furthermore, D and F are the determinants of 2N 2N matrice [akl] and [ekl], respectively; the coefficients Ai and Bi are defined as determinants of a matrix similar to the coefficient matrix [akl], in which the (2i 1)th column or 2ith column is replaced with the constant vector {ck}, respectively. Similarly, the coefficients A0i and B0i are defined as determinants of a matrix similar to the coefficient matrix [ekl], in which the (2i 1)th column or 2ith column is replaced with the constant vector {ck}, respectively. The nonzero elements of the coefficient matrices [akl] and [ekl] and the constant vector {ck} are given as
Y. Ootao, M. Ishihara / Applied Mathematical Modelling 36 (2012) 1431–1443
Þ þ Ha j0 ðb1 la Þ; a1;1 ¼ b1 lj1 ðb1 la
Þ þ Ha y0 ðb1 la Þ; a1;2 ¼ b1 ly1 ðb1 la
a2N;2N1 ¼ Hb j0 ðbN lÞ bN lj1 ðbN lÞ; a2i;2i1 ¼ j0 ðbi lRi Þ;
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a2N;2N1 ¼ Hb y0 ðbN lÞ bN ly1 ðbN lÞ;
a2i;2i ¼ y0 ðbi lRi Þ;
a2i;2iþ1 ¼ j0 ðbiþ1 lRi Þ;
a2i;2iþ2 ¼ y0 ðbiþ1 lRi Þ;
ð9Þ
a2iþ1;2i1 ¼ kri bi lj1 ðbi lRi Þ; a2iþ1;2i ¼ kri bi ly1 ðbi lRi Þ;
a2iþ1;2iþ1 ¼ kr;iþ1 biþ1 lj1 ðbiþ1 lRi Þ;
a2iþ1;2iþ2 ¼ kr;iþ1 biþ1 ly1 ðbiþ1 lRi Þ; i ¼ 1; 2; . . . ; N 1 1 Þ; e2N;2N1 ¼ Hb ; e2N;2N ¼ Hb 1 e1;1 ¼ Ha ; e1;2 ¼ 2 ð1 þ Ha a a e2i;2i1 ¼ 1; e2iþ1;2i ¼
kri
e2i;2i ¼
; 2
Ri
c1 ¼ Ha T a ;
1 ; Ri
e2i;2iþ1 ¼ 1;
e2iþ1;2iþ1 ¼
kr;iþ1 R2i
e2i;2iþ2 ¼
1 ; Ri
ð10Þ ð11Þ
ð12Þ
i ¼ 1; 2; . . . ; N 1
;
c2N ¼ Hb T b ;
ð13Þ
In Eq. (8), D0 (lj) is
D0 ð l j Þ ¼
dD dll¼lj
ð14Þ
and lj is the jth positive root of the following transcendental equation:
DðlÞ ¼ 0:
ð15Þ
2.2. Thermoelastic problem We developed the analysis of a multilayered magneto-electro-thermoelastic hollow sphere as a spherically symmetric state. The displacement–strain relations are expressed in dimensionless form as follows:
erri ¼ u ri ;r ;
ehhi ¼ e//i ¼ u ri =r ;
crhi ¼ cr/i ¼ ch/i ¼ 0;
ð16Þ
where the comma denotes partial differentiation with respect to the variable that follows. For the spherical isotropic and linear magneto-electro-thermoelastic materials, the constitutive relations are expressed in dimensionless form as follows:
r rri ¼ C 11i erri þ 2C 12i ehhi bri T i e1i Eri q1i Hri ; r hhi ¼ r //i ¼ C 12i erri þ ðC 22i þ C 23i Þehhi bhi T i e2i Eri q2i Hri ;
ð17Þ
ri ¼ C 11i a ri þ 2C 12i a hi ; b ri þ ðC 22i þ C 23i Þa hi : bhi ¼ C 12i a
ð18Þ
where
The constitutive equations for the electric and the magnetic fields in dimensionless form are given as
H þp 1i Eri þ d 1i T i ; Dri ¼ e1i erri þ 2e2i ehhi þ g 1i ri e þ 2q e þ d E þ l H þm T: B ¼q ri
1i rri
2i hhi
1i ri
1i
ri
1i i
ð19Þ ð20Þ
The relation between the electric field intensity and the electric potential /i in dimensionless form is defined as
i ;r : Eri ¼ /
ð21Þ
The relation between the magnetic field intensity and the magnetic potential wi in dimensionless form is defined as s
i ;r : Hri ¼ w
ð22Þ
The equilibrium equation in the radial direction is expressed in dimensionless form as follows:
r rri ;r þ 2ðr rri r hhi Þ=r ¼ 0:
ð23Þ
If the electric charge density is zero, the equations of electrostatics and magnetostatics are expressed in dimensionless form as follows:
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Dri ;r þ 2Dri =r ¼ 0;
ð24Þ
Bri ;r þ 2Bri =r ¼ 0:
ð25Þ
In Eqs. (16)–(25), the following dimensionless values are introduced:
rkli ðe ; c Þ uir a C kli Þ ¼ kli kli ; u ki ¼ ki ; C kli ¼ kli ; ri ¼ ; ðekli ; c ; a a0 Y 0 T 0 a0 T 0 a0 T 0 b a0 Y0
r kli ¼ Dri ¼
Dri
a0 Y 0 T 0 jd0 j
Bri ¼
;
eki ; Y 0 jd0 j
g 1i ¼
¼ j0 d1i ; d 1i b
1i ¼ p
eki ¼
Hri ¼
Bri jd0 jj0 ; ba0 T 0
g1i Y 0 jd0 j
2
ki ¼ q
;
p1i
a0 Y 0 jd0 j
;
i ¼ /i jd0 j ; / a0 T 0 b qki j0 jd0 j ; b
1i ¼ m
m1i j0 jd0 j ; ba0
i ¼ w
l 1i ¼
wi ; jd0 jj0 a0 Y 0 T 0
l1i j20 jd0 j2 Y 0
Eri ¼
2
b
;
ð26Þ
Eri jd0 j ; a0 T 0
Hri b ; jd0 jj0 a0 Y 0 T 0
where rkli are the stress components; (ekli, ckli) are the strain components; uri is the displacement in the r direction; aki are the coefficients of linear thermal expansion; Ckli are the elastic stiffness constants; Dri is the electric displacement in the r direction; Bri is the magnetic flux density in the r direction; eki are the piezoelectric coefficients; g1i is the dielectric constant; p1i is the pyroelectric constant; qki are the piezomagnetic coefficients; l1i is the magnetic permeability coefficient; d1i is the magnetoelectric coefficient; m1i is the pyromagnetic constant; and a0, Y0, and d0 are typical values of the coefficient of linear thermal expansion, Young’s modulus and piezoelectric modulus, respectively. Substituting Eqs. (16), (21) and (22) into Eqs. (17), (19) and (20), and later into Eqs. (23)–(25), the governing equations of the displacement uri, electric potential /i, and magnetic potential wi in the dimensionless form are written as
i ;r^r þ 2ðe1i e2i Þ/ i ;rr þ 2ðq i ;r r 1 þ q i ;r r 1 1i w 1i q 2i Þw rir 2 þ e1i / ri ;rr þ 2C 11i u ri ;r r1 þ 2ðC 12i C 22i C 23i Þu C 11i u ri b hi ÞT ir1 þ b ri T i ;r ; ¼ 2ðb
ð27Þ
w r1 ¼ p i ;rr 2g i ;r r 1 d e1i u ri ;rr þ 2ðe1i þ e2i Þu rir 2 g ri ;r r 1 þ 2e2i u 1i ðT i ;r þ 2T ir1 Þ; 1i / 1i / 1i i ;r^r 2d1i wi ;r
ð28Þ
/ r1 l i ;rr 2l i ;r r 1 ¼ m 1i u 1i þ q 2i Þu 2i u ri ;r r 1 þ 2q 1i w 1i w 1i ðT i ;r þ 2T ir 1 Þ: ri ;rr þ 2ðq rir2 d q 1i i ;rr 2d1i /i ;r
ð29Þ
If the inner and outer surfaces of the multilayered magneto-electro-thermoelastic hollow sphere are traction free, and the interfaces of each adjoining layer are perfectly bonded, then the boundary conditions of inner and outer surfaces and the conditions of continuity at the interfaces can be represented as follows:
; r rr1 ¼ 0; r ¼ a r ¼ Ri ; r rri ¼ r rr;iþ1 ; r ¼ 1;
ri ¼ u r;iþ1 ; u
i ¼ 1; . . . ; N 1;
ð30Þ
r rrN ¼ 0:
The boundary conditions in the radial direction for the electric and magnetic fields are expressed as
1 ¼ 0; w 1 ¼ 0; Dr1 ¼ 0; Br1 ¼ 0 or / i ¼ / iþ1 ; w i ¼ w iþ1 ; r ¼ Ri ; Dri ¼ Dr;iþ1 ; Bri ¼ Br;iþ1 ; / r ¼ 1; DrN ¼ 0; BrN ¼ 0 or /N ¼ 0; wN ¼ 0:
; r ¼ a
i ¼ 1; . . . ; N 1;
ð31Þ
The solutions of Eqs. (27)–(29) are assumed in the following form:
rci þ u rpi ; ri ¼ u u
i ¼ / ci þ / pi ; /
i ¼ w ci þ w pi w
ð32Þ
In Eq. (32), the first term on the right-hand side gives the homogeneous solution and the second term gives the particular solution. The homogeneous solutions of Eq. (32) can be expressed as follows:
rci ¼ C 1ir 1 þ C 2irki2 þ C 3ir ki3 ; u ci ¼ C 11i ðC 6i þ C 7ir1 þ g C 2ir ki2 þ g C 3ir ki3 Þ; / 1i 2i e1i ci ¼ C 11i ðC 4i þ C 5ir1 þ g C 2ir ki2 þ g C 3ir ki3 Þ; w 3i 4i 1i q
ð33Þ
Y. Ootao, M. Ishihara / Applied Mathematical Modelling 36 (2012) 1431–1443
1435
where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4b1i =b3i ; 2 # bqi ci bei bdi ð c b b c þ b b Þ ; b1i ¼ ai ci þ b2ei bei di qi i ei di i di ci b2di ki2 ; ki3 ¼ "
1
b3i ¼ 1 þ ci þ
ai ¼
ðci bdi Þ2 di ci b2di
;
2ðC 22i þ C 23i C 12i Þ C 11i
C g ci ¼ 11i2 1i ; e1i g 1i ¼ g 2i ¼
;
bei ¼
2e2i ; e1i
bqi ¼
2i 2q ; 1i q
bdi ¼
1i C 11i d ; 1i e1i q
1i C 11i l ; 21i q 2 ki2 þ ki2 þ bei ðki2 þ 1Þ bdi ki2 a3i ;
di ¼
1
ci ki2 ðki2 þ 1Þ 1
ci ki3 ðki3 þ 1Þ
ð34Þ
2 ki3 þ ki3 þ bei ðki3 þ 1Þ bdi ki3 a4i ;
g 3i ¼ a3i =ðki2 þ 1Þ; g 4i ¼ a4i =ðki3 þ 1Þ; ki2 þ 1 a3i ¼ ½ðci bdi Þki2 þ bqi ci bei bdi ; ki2 di ci b2di ki3 þ 1 a4i ¼ ½ðci bdi Þki3 þ bqi ci bei bdi ki3 di ci b2di In Eq. (33), Cki(k = 1, 2, . . . , 7) are unknown constants. We have the following relation:
ai C 1i þ bei C 7i þ bqi C 5i ¼ 0:
ð35Þ
In order to obtain the particular solutions, series expansions of Sphere Bessel functions given in Eq. (8) are used. Eq. (8) can be written in the following way:
T i ðr i ; sÞ ¼
1 X
ain ðsÞr 2n þ bin ðsÞr 2n1 ;
ð36Þ
n¼0
where
2n 1 2 exp l2j s ð1Þn bi lj X A0i ain ðsÞ ¼ d0n þ Ai ; F ð2n þ 1Þ! lj D0 ðlj Þ j¼1 1 2 exp l2j s ð1Þnþ1 ðbi lj Þ2n1 X B0i Bi : bin ðsÞ ¼ d0n þ F ð2nÞ! lj D0 ðlj Þ j¼1
Fig. 1. Variation of temperature change in the radial direction (Case 1, R1 = 0.85).
ð37Þ
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r in the radial direction (Case 1, R1 = 0.85). Fig. 2. Variation of displacement u
rr and (b) normal stress r hh . Fig. 3. Variation of thermal stresses in the radial direction (Case 1, R1 = 0.85): (a) normal stress r
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1437
Fig. 4. Variation of electric potential in the radial direction (Case 1, R1 = 0.85).
Fig. 5. Variation of magnetic potential in the radial direction (Case 1, R1 = 0.85).
pi , and w pi are obtained in the following forms: rpi ; / Here, d0n is the Kronecker delta. The particular solutions u
rip ¼ u
1 X
fani ðsÞr2nþ1 þ fbni ðsÞr 2n ;
n¼0
ip ¼ / ip ¼ w
1 X n¼0 1 X
hani ðsÞr 2nþ1 þ hbni ðsÞr2n þ hc0i ðsÞ ln r ;
ð38Þ
g ani ðsÞr 2nþ1 þ g bni ðsÞr2n þ g c0i ðsÞ ln r :
n¼0
Expressions for fani(s), fbni(s), hani(s), hbni(s), hc0i(s), gani(s), gbni(s), and gc0i(s) in Eq. (38) have been omitted here for brevity. Then, the stress components, electric displacement, and the magnetic flux density can be evaluated from Eq. (33). Details of the solutions are omitted from here for brevity. The unknown constants in the homogeneous solutions are determined so as to satisfy the boundary conditions in (30) and (31). 3. Numerical results In order to illustrate the foregoing analysis, we consider a two-layered hollow sphere composed of piezoelectric and magnetostrictive layers. The piezoelectric layer is made up of BaTiO3, and the magnetostrictive layer is made up of CoFe2O4. Two
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Fig. 6. Variation of temperature change in the radial direction (Case 2, R1 = 0.85).
r in the radial direction (Case 2, R1 = 0.85). Fig. 7. Variation of displacement u
kinds of two-layered hollow sphere are investigated. Case 1 shows the stacking sequence BaTiO3/CoFe2O4 and Case 2 shows the stacking sequence CoFe2O4/BaTiO3. We assume that the outer surface of the two-layered hollow sphere is heated. Then, numerically calculable parameters of the heat condition and shape are presented as follows:
Ha ¼ Hb ¼ 1:0; T a ¼ 0; T b ¼ 1; N ¼ 2; ¼ 0:7; R1 ¼ 0:75; 0:8; 0:85; 0:9; 0:95; b ¼ 0:01m a
ð39Þ
The following are material constants considered for BaTiO3 [20]:
ah ¼ a/ ¼ 15:7 106 1=K; ar ¼ 6:4 106 1=K; C 22 ¼ 166 GPa; C 23 ¼ 77 GPa; C 12 ¼ 78 GPa; C 11 ¼ 162 GPa e2 ¼ 4:4 C=m2 ;
e1 ¼ 18:6 C=m2 ;
p1 ¼ 2 104 C2 =m2 K;
jr ¼ 0:88 106 m2 =s:
g1 ¼ 12:6 109 C2 =Nm2 ;
l1 ¼ 10 106 Ns2 =C2 ; kr ¼ 2:5 W=mK;
ð40Þ
Y. Ootao, M. Ishihara / Applied Mathematical Modelling 36 (2012) 1431–1443
rr and (b) normal stress r hh . Fig. 8. Variation of thermal stresses in the radial direction (Case 2, R1 = 0.85): (a) normal stress r
Fig. 9. Variation of electric potential in the radial direction (Case 2, R1 = 0.85).
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Fig. 10. Variation of magnetic potential in the radial direction (Case 2, R1 = 0.85).
rr and (b) normal stress r hh . Fig. 11. Variation of thermal stresses in the radial direction (Case 1, s = 1): (a) normal stress r
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Fig. 12. Variation of electric potential in the radial direction (Case 1, s = 1).
Fig. 13. Variation of magnetic potential in the radial direction (Case 1, s = 1).
The corresponding constants for CoFe2O4 are
ar ¼ ah ¼ a/ ¼ 10 106 1=K; C 22 ¼ 286 GPa; C 23 ¼ 173 GPa; C 12 ¼ 170:5 GPa; C 11 ¼ 269:5 GPa; q2 ¼ 580:3 N=Am; q1 ¼ 699:7 N=Am; g1 ¼ 0:093 109 C2 =Nm2 ; 4
p1 ¼ 2 10
2
2
6
2
C =m K;
jr ¼ 0:77 10
l1 ¼ 157 10
6
2
2
Ns =C ;
ð41Þ
kr ¼ 3:2 W=mK;
m =s:
The typical values of material parameters such as j0, k0, a0, Y0, and d0, used to normalize the numerical data, based on those of BaTiO3 are as follows:
j0 ¼ jr ; k0 ¼ kr ; a0 ¼ ar ; Y 0 ¼ 116 GPa; d0 ¼ 78 1012 C=N:
ð42Þ
In the numerical calculations, the boundary conditions at the surfaces for the electric and magnetic fields are expressed as
; r ¼ a r ¼ 1;
Dr1 ¼ 0; Br1 ¼ 0; N ¼ 0: N ¼ 0; w /
ð43Þ
The numerical results for Case 1 and R1 = 0.85 are shown in Figs. 1–5. Fig. 1 shows the variation of temperature change along r along the radial direction. From Figs. 1 and 2, it is clear that the radial direction. Fig. 2 shows the variation of displacement u the temperature and displacement increase with time and have the largest values in steady state. Fig. 3(a) and (b) shows
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rr and r hh , respectively, along the radial direction. Fig. 3(a) reveals that the maximum tensile variations of thermal stresses r stress occurs in the transient state and the maximum compressive stress occurs in the steady state. From Fig. 3(b), it is clear that the compressive stress occurs in the first layer and tensile stress occurs in the second layer. Figs. 4 and 5 show variations and magnetic potential w, respectively, along the radial direction. Fig. 4 reveals that the absolute value of electric potential / of the electric potential increases with time, except during the early stage of heating, and attains its maximum value in the steady state. The electric potential is almost zero in the second layer, i.e. the magnetostrictive layer. From Fig. 5, it is clear that the absolute value of the magnetic potential increases with time and attains its maximum value in the steady state. The magnetic potential is almost constant in the first layer, i.e. the piezoelectric layer. The numerical results for Case 2 and R1 = 0.85 are shown in Figs. 6–10. Fig. 6 shows the variation of temperature change r along the radial direction. From Figs. 1, 2, 6 and 7, it is along the radial direction. Fig. 7 shows the variation of displacement u r of Case 1 are larger than those of Case 2. Fig. 8(a) and (b) shows the clear that the temperature increase and displacement u rr and r hh , respectively, along the radial direction. From Fig. 8(a), it is clear that the maximum variations of thermal stresses r tensile stress occurs in the steady state. Fig. 8(b) reveals that tensile stress occurs in the first layer and compressive stress and magnetic potential w, respectively, occurs in the second layer. Figs. 9 and 10 show the variations of electric potential / along the radial direction. From these figures, it is clear that the absolute values of the electric and the magnetic potential increases with time and attain their maximum value in the steady state. The electric potential is almost constant in the first layer, i.e. the magnetostrictive layer. In contrast, the magnetic potential is almost zero in the second layer, i.e. the piezoelectric layer. In order to assess the influence of the position of the interface between both layers, numerical results for Case 1 and R1 = 0.75, 0.8, 0.85, 0.9, 0.95 were obtained; these results are shown in Figs. 11–13. Fig. 11(a) and (b) shows the variations rr and r hh , respectively, in the steady state. Figs. 12 and 13 show the variations of electric and magnetic of thermal stresses r rr potential, respectively, in the steady state. From Fig. 11(a) and (b), it is clear that the distribution of the thermal stress r hh increases with an increase changes substantially with a change in the parameter R1 whereas the maximum tensile stress r in R1. It can be seen from Figs. 12 and 13 that the absolute values of electric potential increase and those of magnetic potential decrease with an increase in R1. 4. Conclusion In this study, we obtained the exact solution of the transient thermal stress problem of a multilayered magnetoelectro-thermoelastic hollow sphere under uniform surface heating as a spherically symmetric state. As an illustration, we carried out numerical calculations for a two-layered hollow sphere composed of piezoelectric and magnetostrictive materials and examined its behavior in the transient state in terms of temperature change, displacement, stress, and electric and magnetic potential distributions. Furthermore, the effects of the stacking sequence and position of the interface were investigated. Though numerical calculation were carried out for a two-layered hollow sphere, numerical calculation for multilayered magneto-electro-thermoelastic hollow sphere with an arbitrary number of layer and arbitrary stacking sequence can be carried out. 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