Transient thermal stresses of angle-ply laminated cylindrical panel due to nonuniform heat supply in the circumferential direction

Composite Structures 55 (2002) 95±103

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Transient thermal stresses of angle-ply laminated cylindrical panel due to nonuniform heat supply in the circumferential direction Yoshihiro Ootao *, Yoshinobu Tanigawa Department of Mechanical Systems Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai 599-8531, Japan

Abstract This paper is concerned with the theoretical treatment of transient thermal stress problem involving an angle-ply laminated cylindrical panel consisting of an oblique pile of layers having orthotropic material properties due to nonuniform heat supply in the circumferential direction. We obtain the exact solution for the two-dimensional temperature change in a transient state, and thermal stresses of a simply supported cylindrical panel under the state of generalized plane deformation. As an example, numerical calculations are carried out for a 2-layered angle-ply laminated cylindrical panel, and some numerical results for the temperature change, the displacement and the stress distributions are shown in ®gures. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Elasticity; Thermal stress; Angle-ply laminate; Cylindrical panel; Generalized plane deformation; Transient state

1. Introduction Anisotropic laminated composite materials have been used in various industrial ®elds as structure materials and an increase of their use under a high temperature environment is predicted from now on. One of the causes for damage in these laminated composite materials includes delamination. In order to evaluate this phenomenon, the thermal stress analysis taking into account the transverse shearing stresses and the normal stress in the thickness direction is necessary. In addition, a transient thermal stress analysis as well as a steady thermal stress analysis becomes important, because maximum thermal stress distribution occurs in a transient state which lasts from the beginning of the heating to the steady state. Therefore, we recently analyzed exactly the three-dimensional transient thermal stress problem of cross-ply laminated rectangular plate due to partial heating [1] and the transient thermal stress problem of angle-ply laminated strip due to nonuniform heat supply in the width direction [2] taking into account all transverse stress components. However, these studies discuss the problem in rectangular coordinates. On the other hand, though there are several exact analyses for the isothermal problems of anisotropic *

Corresponding author. Tel.: +81-722-54-9210; fax: +81-722-549904. E-mail address: [email protected] (Y. Ootao).

laminated cylindrical panel [3±7], there are only a few exact analyses concerned with the thermoelastic problems of anisotropic laminated cylindrical panel taking into account transverse stress components. For example, Huang and Tauchert treated exactly a cross-ply cylindrical panel [8] and a doubly curved cross-ply laminate [9] with simply supported edges as a three-dimensional thermoelastic problem. Zenkour and Fares [10] analyzed the thermoelastic problem of composite laminated cylindrical panel using a re®ned ®rst-order theory under several boundary conditions. These papers, however, treated only the thermal stress problems under the steady state temperature distribution that is linear with respect to the thickness direction. To the author's knowledge, the exact analysis for a transient thermal stress problem of anisotropic laminated panel under two-dimensional temperature distribution has not been reported, although there are several analytical studies on anisotropic laminated shell [11±14]. These papers, however, treated only the thermal stress problems under one-dimensional temperature distribution with respect to the radial direction. In the present article, we consider an angle-ply laminated cylindrical panel with simply supported edges due to a nonuniform heat supply in the circumferential direction. We analyze exactly the transient thermal stress and thermal deformation of the laminated cylindrical panel as a generalized plane deformation problem taking into account all transverse stress components.

0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 3 2 - 5

96

Y. Ootao, Y. Tanigawa / Composite Structures 55 (2002) 95±103

2. Analysis We consider an in®nitely long, angle-ply laminated cylindrical panel composed of N layers as shown in Fig. 1, the angular length of the side in the circumferential direction of which is denoted by h0 . The panel's inner and outer radii are designated a and b, respectively. Throughout this article, the index i …i ˆ 1; 2; . . . ; N † is associated with the ith layer of a laminated cylindrical panel from the inner side. It is assumed that each layer maintains the orthotropic material properties and the ®ber direction in the ith layer is alternated with ply angle /i to the z-axis. 2.1. Heat conduction problem We assume that the laminated cylindrical panel is initially at zero temperature and is suddenly heated from the inner and outer surfaces by surrounding media with the relative heat transfer coecients ha and hb . We denote the temperatures of the surrounding media by the functions Ta fa …h† and Tb fb …h† and assume its end surfaces …h ˆ 0; h0 † are held at zero temperature. Then the temperature distribution shows a two-dimensional distribution in r±h plane, and the transient heat conduction equation for the ith layer and the initial and thermal

boundary conditions in dimensionless form are taken in the following forms:  2  oT i o T i 1 oT i jhi o2 T i ˆ jri ; ‡ ‡ os oq2 q oq q2 oh2 i ˆ 1; . . . ; N ; …1† sˆ0:

T i ˆ 0; i ˆ 1; . . . ; N ;

oT 1 Ha T 1 ˆ Ha T a fa …h†; oq q ˆ Ri : T i ˆ T i‡1 ; i ˆ 1; . . . ; …N 1†; qˆa:

q ˆ Ri :

kri

oT i oT i‡1 ˆ kr;i‡1 ; i ˆ 1; . . . ; …N oq oq

oT N ‡ Hb T N ˆ Hb T b fb …h†; oq h ˆ 0; h0 : T i ˆ 0; i ˆ 1; . . . ; N ; qˆ1:

…2† …3† …4† 1†; …5† …6† …7†

where jri ˆ jTi ; jhi ˆ jLi sin2 /i ‡ jTi cos2 /i ;

…8†

kri ˆ kTi : In expressions (1)±(8), we have introduced the following dimensionless values: …Ti ; Ta ; Tb † ; T0 …r; ri ; a† ; …q; Ri ; a† ˆ b …jri ; jhi ; jLi ; jTi † …jri ; jhi ; jLi ; jTi † ˆ ; j0 kri kri ˆ ; k0 j0 t sˆ 2 ; b …Ha ; Hb † ˆ …ha ; hb †b; …T i ; T a ; T b † ˆ

…9†

where Ti is the temperature change of the ith layer; jri and jhi are the thermal di€usivities in the r and h directions, respectively; kri is the thermal conductivity in the r direction; t is time; T0 ; j0 , and k0 are the typical values of temperature, thermal di€usivity and thermal conductivity, respectively. In Eq. (8), the subscripts L and T denote the ®ber and transverse directions, respectively. Moreover, ri …i ˆ 1; 2; . . . ; N 1† are the coordinates of interface of the laminated cylindrical panel. Introducing the ®nite sine transformation with respect to the variable h and Laplace transformation with respect to the variable s, the solution of Eq. (1) can be obtained so as to satisfy conditions (2)±(7). This solution is shown as follows:

Fig. 1. Analytical model and coordinate system.

Ti ˆ

1 X kˆ1

T ik …q; s† sin qk h;

i ˆ 1; . . . ; N ;

…10†

Y. Ootao, Y. Tanigawa / Composite Structures 55 (2002) 95±103

where T ik …q; s† ˆ

"

2 1 0 ci 0 …A q ‡ Bi q h0 F i

ci

†‡

D0 …lj † ˆ

1 X 2 exp… l2j s† lj D0 …lj † jˆ1 #

 fAi Jci …bi lj q† ‡ Bi Yci …bi lj q†g ;

…11†

where Jc … † and Yc … † are the Bessel functions of the ®rst and second kind of order c , respectively; D and F are the determinants of 2N  2N matrices ‰akl Š and ‰ekl Š, respectively; the coecients Ai and Bi are de®ned as the determinant of the matrix similar to the coecient matrix ‰akl Š, in which the …2i 1†th column or 2ith column is replaced by the constant vector fck g, respectively; 0 0 similarly, the coecients Ai and Bi are de®ned as the determinant of the matrix similar to the coecient matrix ‰ekl Š, in which the …2i 1†th column or 2ith column is replaced by the constant vector fck g, respectively. Furthermore, the nonzero elements akl and ck of the coecient matrix ‰akl Š and the constant vector fck g are given as follows:  c1  a11 ˆ b1 lJc1 ‡1 …b1 la† ‡ Ha Jc1 …b1 la†; a  c1  a12 ˆ b1 lYc1 ‡1 …b1 la† ‡ Ha Yc1 …b1 la†; a a2N ;2N 1 ˆ …Hb ‡ cN †JcN …bN l† bN lJcN ‡1 …bN l†; a2N ;2N ˆ …Hb ‡ cN †YcN …bN l† a2i;2i 1 ˆ Jci …bi lRi †;

bN lYcN ‡1 …bN l†;

a2i;2i ˆ Yci …bi lRi †; a2i;2i‡1 ˆ a2i;2i‡2 ˆ

Jci‡1 …bi‡1 lRi †; Yci‡1 …bi‡1 lRi †;   c a2i‡1;2i 1 ˆ kri i Jci …bi lRi † bi lJci ‡1 …bi lRi † ; Ri   ci Yci …bi lRi † bi lYci ‡1 …bi lRi † ; a2i‡1;2i ˆ kri Ri  c a2i‡1;2i‡1 ˆ kr;i‡1 i‡1 Jci‡1 …bi‡1 lRi † Ri  bi‡1 lJci‡1 ‡1 …bi‡1 lRi † ;  c a2i‡1;2i‡2 ˆ kr;i‡1 i‡1 Yci‡1 …bi‡1 lRi † Ri  bi‡1 lYci‡1 ‡1 …bi‡1 lRi † ; i ˆ 1; . . . ; …N

c1 ˆ Ha T a f^a …q†;

…12†

1†; c2N ˆ Hb T b f^b …q†:

dD kp ; qk ˆ ; dl lˆlj h0 r jhi 1 qk ; bi ˆ p ; ci ˆ jri jri

97

…13†

On the other hand, the element ekl of the coecient matrix ‰ekl Š is omitted here for the sake of brevity. In Eq. (13), q represents the parameter of ®nite sine transformation with respect to the variable h and a symbol …^ † represents the image function. In Eqs. (10) and (11), D0 …lj †; qk ; bi and ci are

…14†

and lj represents the jth positive roots of the following transcendental equation: D…l† ˆ 0:

…15†

2.2. Thermal stress analysis We now analyze the transient thermal stress of an angle-ply laminated cylindrical panel with simply supported edges as a generalized plane deformation problem. In each layer of the laminated cylindrical panel, the ®ber direction, the in-plane transverse direction and the radial direction are denoted by L, T and R, respectively. Each layer has orthotropic material properties between the ®ber-reinforced direction and its orthogonal direction. Stress±strain relations in dimensionless form for the ith layer are given by the following relations: 3 9 2 8 rLLi > Q11i Q12i Q12i 0 0 0 > > > 6Q >r > > 0 0 0 7 TTi > > 6 12i Q22i Q23i > 7 > = 6 7 0 0 0 7 6 Q12i Q23i Q22i RRi ˆ6 7 > rTRi > 6 0 0 0 Q44i 0 0 7 > > > > 7 6 > > > 4 0 > 0 0 0 Q55i 0 5 > ; : rRLi > rLTi 0 09 0 0 Q66i 80 e a T > > LLi Li i > > > > > > > eTTi aTi T i > > > > > = < eRRi aTi T i : …16†  > > eTRi > > > > > > > > eRLi > > > > ; : eLTi Applying the coordinate transformation rule to Eq. (16), stress±strain relations for the global coordinate system …r; h; z† are    3 9 2 Q 8 Q12i Q13i 0 0 Q16i rzzi > 11i > > >    7 > 6  > > 0 0 Q26i 7 > > 6 Q12i Q22i Q23i rhhi > > > > > 7 6 >    = 6 Q 0 0 Q36i 7 rri 7 6 13i Q23i Q33i ˆ6 7   > > 6 r 0 0 Q44i Q45i 0 7 > 6 0 > rhi > > 7 > > > 7   > 6 > rrzi > > > 5 4 0 0 0 Q Q 0 > > 45i 55i ; :     rhzi Q16i Q26i Q36i 0 0 Q66i 9 8 e a T zzi zi i > > > > > > > > > e a hhi hi T i > > > > > > > > crhi > > > > > > > > > > crzi > > > > ; : chzi ahzi T i

98

Y. Ootao, Y. Tanigawa / Composite Structures 55 (2002) 95±103

where  Q11i  Q12i  Q13i  Q16i 

ˆ ˆ ˆ ˆ

Q22i ˆ  Q23i ˆ  Q26i ˆ 

Q36i  Q44i  Q45i  Q55i  Q66i 

Q33i

m4i Q11i ‡ 2m2i n2i …Q12i ‡ Q66i † m4i Q12i ‡ m2i n2i …Q11i ‡ Q22i ‡ n4i Q12i ; m2i Q12i ‡ n2i Q23i ; mi ni fm2i …Q11i Q12i Q66i † ‡ n2i …Q12i Q22i ‡ Q66i †g; m4i Q22i ‡ 2m2i n2i …Q12i ‡ Q66i † n2i Q12i ‡ m2i Q23i ; mi ni fm2i …Q12i Q22i ‡ Q66i † ‡ n2i …Q11i Q12i Q66i †g;

ˆ mi ni …Q12i Q23i †; ˆ …m2i Q44i ‡ n2i Q55i †=2; ˆ mi ni …Q55i Q44i †=2; ˆ …n2i Q44i ‡ m2i Q55i †=2; ˆ m2i n2i …Q11i ‡ Q22i 2Q12i ‡ …m4i ‡ n4i †Q66i =2; ˆ Q22i ;

‡ 2Q66i †





…Q22i ‡ Q44i †q 2 uhi;h







Q26i q 2 uzi;h







bhi †q 1 T i ;



…23†





…Q44i ‡ Q23i †q 1 uri;qh ‡ …Q44i ‡ Q22i †q 2 uri;h

‡ n4i Q11i ;



…18†



‡ Q44i …q 1 uhi;q

q 2 uhi ‡ uhi;qq † ‡ Q22i q 2 uhi;hh





‡ Q45i …uzi;qq ‡ 2q 1 uzi;q † ‡ Q26i q 2 uzi;hh ˆ q 1 bhi T i;h ;  …Q36i

‡ ‡

Q66i †

‡

…24†

 Q45i †q 1 uri;qh

 Q26i q 2 uhi;hh  Q66i q 2 uzi;hh

‡

‡

 Q26i q 2 uri;h

 Q55i …uzi;qq

‡

 Q45i uhi;qq

‡ q 1 uzi;q †

ˆ bhzi q 1 T i;h ;

…25†

where 































bzi ˆ Q11i azi ‡ Q12i ahi ‡ Q13i ari ‡ Q16i ahzi ; …19†

bhi ˆ Q12i azi ‡ Q22i ahi ‡ Q23i ari ‡ Q26i ahzi ; bri ˆ Q13i azi ‡ Q23i ahi ‡ Q33i ari ‡ Q36i ahzi ;

where rkli are the stress components, ekli are the strain tensor, ckli are the engineering shear strain components, aki and ahzi are the coecients of linear thermal expansion, Qkli and Qkli are the elastic sti€ness constants, and a0 and E0 are the typical values of the coecient of linear thermal expansion and Young's modulus of elasticity, respectively. Next, we assume the displacement components for the global coordinate system in the following forms: uzi ˆ uzi …q; h†;

…21†

where uri ; uhi and uzi are the dimensionless quantities of displacements in the r; h and z directions, respectively, and the dimensionless values introduced in Eq. (21) are given by the following relations: …uri ; uhi ; uzi † ˆ …uri ; uhi ; uzi †=a0 T0 b:

Q44i uri;hh †



ˆ bri T i;q ‡ …bri

ni ˆ sin /i :

uhi ˆ uhi …q; h†;



q 2 …Q22i uri

‡ …Q36i ‡ Q45i †q 1 uzi;qh

In expressions (16)±(19), the following dimensionless values are introduced: rkli rkli ˆ ; a 0 E0 T 0 …ekli ; ckli † ; …ekli ; ckli † ˆ a0 T 0 …20† …aki ; ahzi † ; …aki ; ahzi † ˆ a0 …Qkli ; Qkli †  ; …Qkli ; Qkli † ˆ E0

uri ˆ uri …q; h†;



Q33i …uri;qq ‡ q 1 uri;q †

‡ …Q23i ‡ Q44i †q 1 uhi;qh

azi ˆ m2i aLi ‡ n2i aTi ; ahi ˆ n2i aLi ‡ m2i aTi ; ari ˆ aTi ; ahzi ˆ 2mi ni …aLi aTi †; mi ˆ cos /i ;

equilibrium equations, the displacement equations of equilibrium are written as:

n4i Q22i ;

…22†

Taking into account Eq. (21) and substituting the displacement±strain relations and Eq. (17) into the

…26†

bhzi ˆ Q16i azi ‡ Q26i ahi ‡ Q36i ari ‡ Q66i ahzi : In Eqs. (23)±(25), a comma denotes partial di€erentiation with respect to the variable that follows. The boundary conditions of inner and outer surfaces and the conditions of continuity on the interfaces can be represented as follows: qˆa: qˆ1: q ˆ Ri :

rrr1 ˆ 0; rrh1 ˆ 0; rrz1 ˆ 0; rrrN ˆ 0; rrhN ˆ 0; rrzN ˆ 0; rrri ˆ rrr;i‡1 ; rrhi ˆ rrh;i‡1 ; rrzi ˆ rrz;i‡1 ; uri ˆ ur;i‡1 ; uhi ˆ uh;i‡1 ; uzi ˆ uz;i‡1 :

…27†

We now consider the case of a simply supported panel given by the following relations: h ˆ 0; h0 :

rhhi ˆ 0;

rhzi ˆ 0;

uri ˆ 0:

…28†

We assume the solutions of Eqs. (23)±(25) in order to satisfy Eq. (28) in the following form: 1 X uri ˆ fUrcik …q† ‡ Urpik …q†g sin qk h; kˆ1

1 X uhi ˆ fUhcik …q† ‡ Uhpik …q†g cos qk h;

uzi ˆ

…29†

kˆ1 1 X

fUzcik …q† ‡ Uzpik …q†g cos qk h:

kˆ1

In expressions (29), the ®rst term on the right-hand side gives the homogeneous solution and the second term on the right-hand side gives the particular solution. We

Y. Ootao, Y. Tanigawa / Composite Structures 55 (2002) 95±103

99

now consider the homogeneous solution and introduce the following equation:

We now introduce the following expression:

q ˆ exp…s†:

Hi ˆ

…30†

Substituting the ®rst term on the right-hand side of Eq. (29) into the homogeneous expression of Eqs. (23)±(25), and later changing a variable with the use of Eq. (30), we have 



2



‰Q33i D …Q22i ‡ Q44i q2k †ŠUrcik     ‰…Q23i ‡ Q44i †D …Q22i ‡ Q44i †Šqk Uhcik   ‡ ‰Q26i …Q36i ‡ Q45i †DŠqk Uzcik ˆ 0;     ‰…Q44i ‡ Q23i †D ‡ …Q44i ‡ Q22i †Šqk Urcik    2 ‡ ‰Q44i D …Q44i ‡ Q22i q2k †ŠUhcik   ‡ ‰Q45i D…D ‡ 1† Q26i q2k †ŠUzcik ˆ    ‰Q26i ‡ …Q36i ‡ Q45i †DŠqk Urcik   ‡ ‰Q45i D…D 1† Q26i q2k ŠUhcik   2 ‡ ‰Q55i D Q66i q2k ŠUzcik ˆ 0;

…31†

fi2 di3 ‡ : 4 27

From Eq. (36), there might be three distinct real roots; three real roots with at least two of them being equal or a real root in conjunction with one pair of conjugate complex roots depending on Hi being negative, zero or positive, respectively. For instance, Urcik …q†; Uhcik …q† and Uzcik …q† can be expressed as follows when Hi < 0: 3 X

Urcik …q† ˆ

0;

Uhcik …q† ˆ

3 X

Uzcik …q† ˆ

J ˆ1

where d : ds We express Urcik ; Uhcik and Uzcik as follows:

…34†

0 0 0 …Urcik ; Uhcik ; Uzcik † ˆ …Urcik ; Uhcik ; Uzcik † exp…ki s†:

…35†

Substituting Eq. (35) into Eqs. (31)±(33), the condition 0 0 0 that nontrivial solutions of …Urcik ; Uhcik ; Uzcik † exist leads to the following equation: pi3 ‡ di pi ‡ fi ˆ 0;

…36†

where pi ˆ

k2i "

fi ˆ

3 X J ˆ1

Dˆ

B…i† ; 3A…i†

" di ˆ

3A…i† C …i† ‡ …B…i† †2

3

3…A…i† †

2…B…i† † ‡ 9A…i† B…i† C …i† ‡ 27D…i† …A…i† † 27…A…i† †3

…39†

J Uzcik …q†;

where …i†

…i†

J Urcik …q† ˆ FrJ qmJi ‡ SrJ q J Uhcik …q†

ˆ

…i† LkiJ …mJi †FrJ qmJi

mJi

; …i†

‡ LkiJ … mJi †SrJ q

…i†

mJi

; …40†

…i†

J …q† ˆ RkiJ …mJi †FrJ qmJi ‡ RkiJ … mJi †SrJ q mJi ; Uzcik r B…i† B…i† mJi ˆ piJ ‡ …i† if piJ ‡ …i† > 0; 3A 3A J Urcik …q† ˆ FriJ cos…mJi ln q† ‡ SriJ sin…mJi ln q†;

…i†

Im‰LkiJ …jmJi †Š sin…mJi ln q†gFrJ

; 2

J Uhcik …q†;

J Uhcik …q† ˆ fRe‰LkiJ …jmJi †Š cos…mJi ln q†

#

2

J Urcik …q†;

J ˆ1

…32†

…33†

…38†

‡ fIm‰LkiJ …jmJi †Š cos…mJi ln q†

#

    2 A ˆ Q33i ‰Q55i Q44i …Q45i † Š; 2 B…i† ˆ q2k Q55i ‰Q33i Q22i …Q23i † Š 2 ‡ q2k Q44i ‰Q33i Q66i …Q36i † Š    ‡ …Q33i ‡ Q22i 2q2k Q23i †    2  ‰Q55i Q44i …Q45i † Š      2q2k Q45i …Q33i Q26i Q23i Q36i †;   C …i† ˆ ‰q4k Q36i ‡ 2q2k …q2k 1†Q45i Š      …Q22i Q36i Q23i Q26i †     2 q4k Q33i ‰…Q22i Q66i …Q26i † Š       ‡ q4k …Q23i ‡ 2Q44i †…Q23i Q66i Q36i Q26i †     Q22i …q2k 1†2 ‰Q55i Q44i …Q45i †2 Š    ‡ q2k Q44i ‰…Q36i †2 ‡ …Q26i †2     Q33i Q66i Q22i Q66i Š; 2 2 D…i† ˆ q2k …q2k 1† Q44i ‰Q22i Q66i …Q26i † Š:

…i†

‡ Re‰LkiJ …jmJi †Š sin…mJi ln q†gSrJ ;

;

J Uzcik …q† ˆ fRe‰RkiJ …jmJi †Š cos…mJi ln q†

…41†

…i† Im‰RkiJ …jmJi †Š sin…mJi ln q†gFrJ

…i†

‡ fIm‰RkiJ …jmJi †Š cos…mJi ln q† …i†

…37†

‡ Re‰RkiJ …jmJi †Š sin…mJi ln q†gSrJ ; s   B…i† B…i† mJi ˆ piJ ‡ …i† if piJ ‡ …i† < 0: 3A 3A In Eqs. (40) and (41), 1  Q33i Q45i …x4 ‡ x3 † LkiJ …x† ˆ IkiJ …x† ‡ ‰Q33i Q26i q2k ‡ Q45i …Q22i ‡ q2k Q44i † 







q2k …Q36i ‡ Q45i †…Q44i ‡ Q23i †Šx2 











‡ ‰Q45i …Q22i ‡ q2k Q44i † ‡ q2k Q26i …Q44i ‡ Q23i † 







q2k …Q22i ‡ Q44i †…Q36i ‡ Q45i †Šx   ‡Q26i Q44i q2k …1 q2k † ;

100

Y. Ootao, Y. Tanigawa / Composite Structures 55 (2002) 95±103

RkiJ …x† ˆ

3. Numerical results

1    4 2 Q Q x ‡ ‰q2k …Q23i ‡ Q44i † IkiJ …x† 33i 44i

Q33i …q2k Q22i ‡ Q44i † Q44i …Q22i ‡ q2k Q44i †Šx2 o 2 ‡ Q22i Q44i …q2k 1† ;      IkiJ …x† ˆ qk …Q36i Q44i Q23i Q45i †x3 



‡ …Q22i Q45i





Q23i Q45i





Q26i Q44i †x2

     ‡ ‰q2k Q26i …Q23i ‡ Q44i † ‡ Q45i …Q22i     …Q36i ‡ Q45i †…q2k Q22i ‡ Q44i †Šx   ‡Q26i Q44i …1 q2k † :

‡

 Q44i †

n 2n‡c 1 X … 1† …x=2† ; n!C…c ‡ n ‡ 1† nˆ0

Yc …x† ˆ

1 ‰cos cpJc …x† sin cp

jL ˆ 41:1  10

6

m2 =s;

jT ˆ 29:5  10

6

m2 =s;

aL ˆ 7:6  10 …42†

In Eq. j, Re‰ Š and Im‰ Š are the imaginary unit p(41),  jˆ 1, real part and imaginary part, respectively. …i† …i† Furthermore, in Eqs. (40) and (41), FrJ and SrJ are the unknown constants. The case of Hi ˆ 0 or Hi > 0 is omitted here for the sake of brevity. In order to obtain the particular solution, we use the series expansions of the Bessel functions as follows: Jc …x† ˆ

To illustrate the foregoing analysis, we consider the angle-ply laminated cylindrical panel composed of alumina ®ber-reinforced aluminum composite, with the following properties:

6

aT ˆ 14:0  10

1=K; 6

1=K;

kL ˆ 105 W=m K; kT ˆ 75 W=m K; EL ˆ 150 GPa; ET ˆ 110 GPa;

…47†

GLT ˆ 35 GPa; GTT ˆ 41 GPa; vLT ˆ 0:33; vTT ˆ 0:33; vTL ˆ 0:242; where G and v are the shear modulus of elasticity and Poisson's ratio, respectively. We assume that each layer of laminated cylindrical panel consists of the same or-

…43† J c …x†Š

if c 6ˆ integer: …44†

Since the order ci of the Bessel function in Eq. (11) is not an integer in general except /i ˆ 0°, Eq. (11) can be written as the following expression using Eqs. (43) and (44): T ik …q; s† ˆ

1 X fani …s†q2n‡ci ‡ bni …s†q2n

ci

g:

…45†

nˆ0

Expressions for the functions ani …s† and bni …s† in Eq. (45) are omitted here for the sake of brevity. We assume Urpik …q†; Uhpik …q† and Uzpik …q† as follows: Urpik …q† ˆ Uhpik …q† ˆ Uzpik …q† ˆ

1 X nˆ0 1 X nˆ0 1 X

fani q2n‡ci ‡1 ‡ fbni q2n

ci ‡1




gani q2n‡ci ‡1 ‡ gbni q2n

ci ‡1

hani q2n‡ci ‡1 ‡ hbni q2n

ci ‡1

;







;

…46†

:

nˆ0

Substituting Eqs. (10), (45) and (46) and the second term on the right-hand side of Eq. (29) into Eqs. (23)±(25), and later comparing the coecients of functions with regard to q, respectively, the constants fani ; fbni ; gani ; gbni ; hani and hbni …i ˆ 1; . . . ; N † can be obtained. Then, the stress components can be evaluated by substituting Eq. (29) into the displacement±strain relations, and later into the stress±strain relations. The unknown constants in Eqs. (40) and (41) are determined so as to satisfy the boundary condition (27).

Fig. 2. Temperature change: (a) variation in the heated surface …q ˆ 1†; (b) variation in the radial direction …h ˆ h0 =2†.

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thotropic material, and consider a 2-layered anti-symmetric angle-ply laminated cylindrical panel with the ®ber orientation …/0 = /0 † and the same thickness. We assume that the laminated cylindrical panel is heated from the outer surface by surrounding media, the temperature of which is denoted by the symmetric function with respect to the center of the panel …h ˆ h0 =2†. The numerical parameters of heat conduction and shape are presented as follows: Ha ˆ Hb ˆ 5:0; T a ˆ 0; T b ˆ 1; h0 ˆ 90°; a ˆ 0:7; /0 ˆ 60°; ( 1; hb 6 h0 6 hb ; fb …h† ˆ 0; h0 6 hb ; hb 6 h0 ; hb ˆ 15°; Fig. 3. Variation of thermal displacement uh on the heated surface …q ˆ 1†.

Fig. 4. Thermal displacement ur : (a) variation in the heated surface …q ˆ 1†; (b) variation in the radial direction …h ˆ h0 =2†.

h0 ˆ h

…48†

h0 =2:

The typical values of material properties such as j0 ; k0 ; a0 and E0 , used to normalize the numerical data, are based on those of ®ber direction.

Fig. 5. Thermal stress rhh : (a) variation in the heated surface …q ˆ 1†; (b) variation in the radial direction …h ˆ h0 =2†.

102

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Fig. 2 shows the results of temperature change. The variation on the heated surface …q ˆ 1† is shown in Fig. 2(a) and the variation in the radial direction at the midpoint …h ˆ h0 =2† of the panel is shown in Fig. 2(b). As shown in Fig. 2(a), the temperature rise can clearly be seen in the heated region. As shown in Figs. 2(a) and (b), the temperature rises as the time proceeds and is the greatest in a steady state. Figs. 3 and 4 show the variations of thermal displacement uh and ur , respectively. The variations on the heated surface are shown in Figs. 3 and 4(a) and the variation in the radial direction at the midpoint of the panel is shown in Fig. 4(b). As shown in Figs. 3 and 4(a), the value of thermal displacement uh rises as the time proceeds and has a maximum value in the steady state, while the thermal displacement ur shows the maximum value in the transient state. As shown in Fig. 4(b), the displacement ur changes along the radial direction and shows the maximum value on the heated surface. Fig. 5 shows the variation of the normal stress rhh . The variation on the heated surface is shown in Fig. 5(a) and the

variation in the radial direction at the midpoint of the panel is shown in Fig. 5(b). From Fig. 5(a), though the large compressive stress occurs at the heated region in an initial state of heating, its compressive stress decreases with progress of time and the slight tensile stress occurs in the steady state. From Fig. 5(b), discontinuities of stress occur on the interface …q ˆ 0:85†. Fig. 6 shows the variation of the normal stress rrr . The variation on the interface is shown in Fig. 6(a) and the variation in the radial direction at the midpoint of the panel is shown in Fig. 6(b). As shown in Fig. 6, it can be seen that the stress variation becomes signi®cant with the progress of time and maximum tensile stress occurs at the midpoint of the panel and near q ˆ 0:9 inside of panel in a transient state. Fig. 7 shows the variation of the shearing stress rrz . The variation on the interface is shown in Fig. 7(a) and the variation in the radial direction at the edge …h ˆ 30°† of heated region is shown in Fig. 7(b). As shown in Fig. 7(a), the value of the shearing stress rrz rises as the time proceeds and has a maximum value in the steady state. From

Fig. 6. Thermal stress rrr : (a) variation on the interface …q ˆ 0:85†; (b) variation in the radial direction …h ˆ h0 =2†.

Fig. 7. Thermal stress rrz : (a) variation on the interface …q ˆ 0:85†; (b) variation in the radial direction …h ˆ 30°†.

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stresses of an angle-ply laminated cylindrical panel with simply supported edges due to a nonuniform heat supply in the circumferential direction. Numerical calculations were carried out for a 2-layered anti-symmetric angle-ply laminated cylindrical panel composed of alumina ®ber-reinforced aluminum composite. Though numerical calculations were carried out for a 2-layered anti-symmetric angle-ply laminated cylindrical panel which shows a characteristic of ®ber orientation angle clearly, numerical calculations for hybrid laminated cylindrical panel with an arbitrary number of layer and arbitrary ®ber orientation angles can be carried out. Moreover, transverse shearing stresses and normal stress in the radial direction, which are considered to induce delamination on the interface of each layer, are evaluated precisely in a transient state. References

Fig. 8. Thermal stress rrh : (a) distribution in a transient state …s ˆ 0:01†; (b) variation in the radial direction …h ˆ 30°†.

Fig. 7(b), it can be seen that the shearing stress rrz shows the maximum value on the interface. Fig. 8 shows the results of the shearing stress rrh . The distribution in a transient state …s ˆ 0:01† is shown in Fig. 8(a) and the variation in the radial direction at the edge …h ˆ 30°† of the heated region is shown in Fig. 8(b). Since the shearing stress rrh is anti-symmetric with respect to h ˆ 45° under the condition of Eq. (47), Fig. 8(a) shows the range 0±45°. As shown in Fig. 8(a), it can be seen that the maximum stress occurs near h ˆ 30° inside of panel. From Fig. 8(b), it can be seen that the shearing stress rrh shows the maximum value near q ˆ 0:94 in a transient state. 4. Conclusions In the present article, we obtained the exact solution for the transient temperature and transient thermal

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