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A HIGHER ORDER DYNAMIC THEORY FOR ISOTROPIC THERMOELASTIC CYLINDRICAL SHELLS, PART 1: THEORY A.-H. M Department of Civil Engineering

G. A. B Y. M Department of Engineering Science, Middle East Technical University, Ankara, Turkey (Received 20 May 1992, and in final form 20 October 1993) Through the use of a new technique, a higher order approximate theory is developed to predict the dynamic response of isotropic thermoelastic cylindrical shells. The proposed mth order approximate theory takes into account the interaction between the thermal and mechanical effects. The new technique enables one to satisfy the lateral boundary conditions of the cylindrical shells (CS) accurately. This, in turn, results in the correct description of geometric dispersion of thermoelastic waves propagating in CS. The order of the approximate theory is kept arbitrary in the formulation; thus, by increasing it it is possible to improve the prediction of the theory.

1. INTRODUCTION

With the object of developing refined dynamic theories for beams, plates and shells, a new technique was proposed in reference [1]. In the present study, this technique is employed to develop a higher order approximate theory for isotropic cylindrical thermoelastic shells. In the development of the approximate theory, the interactions between the thermal and mechanical effects are taken into account. The material is assumed to be linear. Two types of field variables are introduced in the approximate theory. The first type includes the weighted averages of stresses, displacements, temperature, etc., over the thickness of the shell, which are termed as weighted variables (WV). The second type describes the displacements, stress tractions, temperature and heat flux at the lateral surface of the shell. These are called face variables (FV). Both WVs and FVs are unknowns of the theory. The appearance of FVs in the theory reflects the novelty of the approximate theory, as opposed to the theories proposed in literature (see, e.g., references [2–6]) which contain only WVs. The presence of FVs in the theory enables one to satisfy the lateral boundary conditions, and thus to predict the geometric dispersion characteristics of the shell correctly without introducing any matching coefficients into the approximate theory. Treating FVs as the field variables of the approximate theory is very crucial in developing refined dynamic approximate theories for composite materials. This was verified in reference [7] for layered thermoelastic composites. In that work it was shown that the approximate theory incorporating FVs as field variables is capable of predicting the periodic structure of the spectra for waves propagating perpendicular to the planes of layering, which has not been achieved by other approximate theories. 817 0022–460X/95/050817 + 10 $08.00/0

7 1995 Academic Press Limited

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In this study, the order of the theory is kept arbitrary. The technique employed in the theory, which is based on a modified Galerkin method, involves the selection of a set of weighting functions (WFs). No restriction is imposed on the choice of the WFs other than that they should be complete. The theory contains two types of equations. The first set is obtained by taking the weighted averages of the field equations of thermoelasticity over the thickness of the shell. The second set comprises the so-called constitutive equations of the face variables. They are derived by expanding the displacements and temperature in terms of WFs over the thickness of the shell, and by taking into account the relations between these expansions and the definitions of the WVs and FVs. The values of the constants appearing in the approximate equations developed depend upon the choice of the WFs. In this study, these values are presented when the WFs are Legendre polynomials. The study is organized as follows. In section 2, the exact equations of thermoelastic cylindrical shells are presented. The general form of the approximate theory is developed in section 3. The expressions for the constants appearing in the approximate theory are presented in section 4 for the case in which the WFs are orthogonal. That section also contains the values of these constants when the WFs are Legendre polynomials. Finally, some discussions about the validity of the theory are presented in section 5.

2. THE EQUATIONS OF THERMOELASTICITY FOR CYLINDRICAL SHELLS

The cylindrical shell is assumed to be made of a linear isotropic thermoelastic material having the material constants r, (m, l), a, cv and kh , which are, respectively, the mass density, Lame´’s constants, the coefficient of thermal expansion, the specific heat per unit volume and the coefficient of heat conduction. The shell has uniform thickness 2h (see Figure 1) and is referred to an orthogonal curvilinear co-ordinate system (y1 , y2 , y3 ) in which the y1 –y3 surface coincides with the mid-surface of the shell. y1 is parallel to the axis of the shell and y2 is directed away from the center of curvature C. The radius of curvature R may vary along the y3 -axis. The fundamental equations of thermoelasticity written in the (y1 , y2 , y3 ) curvilinear co-ordinate system described above are as follows: the equations of motion, 1a taj + w(13 t3j + tij Ci3 + t3i Cij ) + fj = ru¨j ;

(1)

the constitutive equations tab = m(1a ub + 1b ua ) + ldab (1a ua + w 13 u3 + wui Ci3 ) − bi udab , ta3 = t3a = m(1a u3 + w 13 ua + wCia ui ), t33 = (2m + l)w(13 u3 + ui Ci3 ) + l 1a ua − bt u;

(2)

H − (1a qa + w 13 q3 + wCi3 qi ) = cn u + bt T0 (1a va + w 13 v3 + wCi3 vi );

(3)

the energy equation,

and the modified Fourier’s law, tq˙a + qa = −kh 1a u,

tq˙3 + q3 = −kh w 13 u.

In equations (1)–(4), a, b = 1, 2 and i, j = 1, 2, 3, bt = (3l + 2m)a,

K0 (Cij ) = G0 k0

0 0 −1/R

0 L 1/RG, 0 l

w = R/(R + y2 ).

(4)

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Figure 1. The geometrical description of the cylindrical shell.

Also in equations (1)–(4), tij , fj , uj , vj and qj , u, H, t denote, respectively, stress, body force, displacement and velocity components, and the heat flux component, the temperature deviation from the reference temperature T0 , the heat generation per unit volume and the retardation time for heat flux; a dot designates the time derivative and 1j denotes 1/1y j . dab appearing in equations (2) is Kronecker’s delta. In writing equations (1)–(3), the summation convention is used.

3. THE APPROXIMATE THEORY

In what follows we develop the general form of the approximate theory. The approximate governing equations consist of the field equations obtained by integrating the equations presented in the previous section over the shell thickness, the constitutive equations for FVs and the lateral boundary conditions. 3.1. To develop the approximate theory, first a set of WF’s {fn (y¯2 ); n = 0, 1, 2, . . .} with y¯2 = y2 /h are chosen. These form a complete set and, without loss of generality, we assume that fn is an even function of y¯2 for even n and odd function of y¯2 for odd n. For an mth order theory, we retain the elements (f0 , f1 , . . . , fm ; fm + 1 , fm + 2 ) of the set. As will be seen later, keeping the last two elements of the set will permit the establishment of constitutive equations for FVs. h We now apply the operator Ln = (1/2h) f−h ( )fn dy2 (n = 0, . . . , m) to equation (1),

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ET AL.

which yields n n 11 t11 + R1n − t¯ 21 + 13 *t n31 − *t¯ n31 + (t*n21 /R) + f 1n = ru¨1n , n n 11 t12 + R2n − t¯ 22 + 13 *t n32 − *t¯ n32 + {(t* n22 − *t n33 )/R} + f 2n = ru¨2n , n n 11 t 13 + R3n − t¯ 23 + 13 *t n33 − *t¯ n33 + (2t* n23 /R) + f 3n = ru¨3n ,

n = 0, . . . , m,

(5)

where * * ,L (t¯ ijn , *t nij , *t¯ nij ) = (L n , L n n )tij ,

(tijn , f in , uin ) = Ln (tij , fi , ui ),

Ln =

1 2h

g

− + − * n = Ri = t2i − t2i R i + − R+ i = t2i + t2i

6

fn (1) * n Ri , 2h

Rin = h

* 1 * ,L (L n n ) = 2h

( ) 12 fn dy2 ,

−h

g

for even n for odd n

7

,

h

( )(wfn , 13 wfn ) dy2 ,

t3 2i = t2i=y 2 = 3h . (6)

−h

To simplify the theory, we will expand f'n = dfn /dy¯2 , wfn and 13 wfn in terms of fj (j = 0, . . . , m) as m

m

f'n = s cnj fj ,

m

wfn = s c¯nj fj ,

j=0

13 wfn = s cn j fj .

j=0

(7)

j=0

When the weighting functions fi have been chosen, the coefficients cnj , c¯nj and cn j can be determined by applying the operator Lk (k = 0, . . . , m) to both sides of equations (7) and solving the resulting equations for the above-mentioned coefficients. In view of equations (6) and the expansions (7), (t¯ ijn , *t mij , *t¯ nij ) can be expressed in terms of tijn as t¯ ijn =

1 m s c tk , h k = 0 nk ij

m

m

*t nij = s c¯nk tijk ,

*t¯ n = s c t k . nk ij ij

k=0

k=0

(8)

To establish constitutive equations for tijn , we apply the operator Ln (n = 0, . . . , m) to equations (2). This gives n t11 = (2m + l) 11 u1n + l(S2n − u¯2n ) + l(13 u*¯ n3 − u*¯ n3 + u* n2 /R) − bt u n, n t22 = (2m + l)(S2n − u¯2n ) + l 11 u1n + l(13 u* n3 − u*¯ n3 + u* n2 /R) − bt u n, n t33 = (2m + l)(13 u* n3 − u*¯ n3 + u* n2 /R) + l 11 u1n + l(S2n − u¯2n ) − bt u n, n t12 = m 11 u2n + m(S1n − u¯1n ),

n t23 = m(S3n − u¯ 3n ) + m(13 u* n2 − u*¯ n2 − u* n3 /R),

n t13 = m(11 u3n + 13 u* n1 − u*¯ n1 ),

n = 0, . . . , m,

(9)

where u n = Ln (u), Sin =

fn (1) * n S , 2h i

* * ,L n , L (u¯ jn , u* nj , u*¯ nj ) = (L n n )uj ,

6

− + − * n = Si = ui − ui S i + − S− i = ui − u i

for even n for odd n

7

u3 i = ui=y 2 = 3h .

,

(10)

By using the expansions given in equations (7), it may be shown that (u¯ jn , u* nj , u*¯ nj ) can be related to ujn by u¯jn =

1 m s c uk , h k = 0 nk j

m

u*nj = s c¯nk ujk , k=0

m

u*¯ nj = s c nk ujk . k=0

(11)

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We now integrate the energy and Fourier’s equations. To this end, we apply the operator Ln (n = 0, . . . , m) to equations (3) and (4), which gives H n−(11 q1n+Q n−q¯2n+13 q* n3−q*¯ n3 + q* n2 /R) = cv u n + bt T0 (11 v1n + S 2n − v¯2n + 13 v3n − *v¯ n3 + *v n2 /R) (12) and tq˙2n + q2n = −kh (c n − un),

tq˙1n + q1n = −kh 11 u n,

*n u ), tq˙3n + q3n = −kh (13 u*n −

n = 0, . . . , m,

(13)

where (vjn , v¯ jn , *v nj , *v¯ nj ) = (1/1t)(ujn , u¯jn , u* nj , u*¯ nj ), *n * * ,L n, u* n, (u u ) = (L n , L n n )u,

(Hn , qjn ) = Ln (H, qj ),

− + − for even n *n = c = u − u c , + + − c =u +u for odd n * * ,L n , L (q¯jn , q* nj , q*¯ nj ) = (L n n )qj ,

cn =

fn (1) * n c, 2h

Qn =

fn (1) * n Q, 2h

6

7

6

7

+ − − * n = Q = q2 − q2 for even n , Q − Q+ = q+ for odd n 2 + q2

u3 = u=y 2 = 3h ,

q3 2 = q2 =y 2 = 3h .

(14) In view of equations (7) and (14), the weighted heat flux components (q¯ , q* , q*¯ nj ) * n , u*n, and the weighted temperatures (u u) appearing in equations (12) and (13) can be n n expressed in terms of (qj and u ) as n j

(q¯jn , un) =

1 m s c (q k , u k), h k = 0 nk j

n j

m

(q*nj , u*n ) = s c¯nk (qjk , u k),

m *n u ) = s c nk (qjk , u k). (q*¯ nj ,

k=0

(15)

k=0

The integration of field equations is now complete. Equations (5), (9), (12) and (13) constitute 13(m + 1) equations for 13(m + 1) + 16 unknowns; namely, mechanical 3 n n 3 3 variables, tijn , ujn , R3 j , Sj and thermal variables, qj , u , Q , c . Here it may be noted that n n n n (tij , uj ) and (qj , u ) are, respectively, the weighted averages of (stresses, displacements) and (flux, temperature). In this study they will be called weighted variables (WVs). On the other 3 3 3 hand, the variables (R3 j , Sj ) and (Q , c ) are defined on the lateral surfaces of the shell. They are therefore, termed face variables (FVs). To complete the formulation, 16 more equations are needed. Eight equations will come from lateral boundary conditions. In fact, at least one member of each pair (t2j , uj , j = 1–3) and (q2 , u) is prescribed on the lateral surfaces of the shell, which results in eight conditions. Through the use of equations (6), 3 3 (10) and (14), these conditions in turn, can be expressed in terms of FVs R3 and j , Sj , Q 3 c . The remaining eight equations will be obtained by establishing the constitutive 3 equations for the FVs R3 j and Q . 3.2. s To establish these equations, we expand the displacements uj and temperature u in terms

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of DFs, fj s, as m+2

m+2

uj = s aij fi

u = s bi fi .

and

i=0

(16)

i=0

Here, aij and bi are some functions of y1 , y3 and t. Constitutive equations (CEs) for FVs are established in two steps: first, the aij and bi n 3 appearing in equations (16) are determined in terms of (uin , S3 i ) and (u , c ), respectively, by applying the operator Ln (n = 0, . . . , m) to both sides of equations (16) and by inserting 3 equations (16) in the definitions of S3 given in equations (10) and (14); then, the i and c j 3 expressions for ai and bi are introduced into the definitions of the FVs R3 j and Q given in 3 3 equations (6) and (14). Following these two steps, CEs for Rj and Q may be obtained as R+ 1 =

m h

0

p'

1

2gk u1k + g−S− + m 11 S+ 1 2 ,

s k = 1,3,...

R+ 2 =

(2m + l) h

0

p'

(2m + l) h

0

m h

0

p

1

2gk u1k + g+S+ + m 11 S− 1 2 ,

s k = 0,2,...

1

2gk u2k + g−S− 2

s k = 1,3,...

+ l 11 S+ 1 + R− 2 =

R− 1 =

l − + − + (ES+ 2 + FS2 ) + l(E 13 S3 + F 13 S3 ) − bt c , R p

1

2gk u2k + g+S+ + l 11 S − 2 1 +

s k = 0,2,...

l + (ES− 2 + FS2 ) R

+ − + l(E 13 S− 3 + F 13 S3 ) − bt c ,

R+ 3 =

R− 3 =

m h

0

m h

0

p'

1

m − + − (ES+ 3 + FS3 ) + m(E 13 S2 + F 13 S2 ), R

1

m − + − (FS+ 3 + ES3 ) + m(F 13 S2 + E 13 S2 ), R

2gk u3k + g−S− − 3

s k = 1,3,...

p

2gk u3k + g+S+ − 3

s k = 0,2,...

(17)

and

0

1 k + 1 Q+ = − h h 1t

0

1 k + 1 Q− = − h 1t h

t

t

1

0

s

1

0

s

p'

k = 1,3,...

p

k = 0,2,...

1

2gk u k + g−c− ,

1

2gk u k + g+c+ ,

(18)

where E = R 2/(R 2 − h 2),

F = −Rh/(R 2 − h 2),

6

p = m and p' = m − 1

for even m

p = m − 1 and p' = m

for odd m

7

,

(19) and R is the radius of curvature. gi and g3 appearing in equations (17) and (18) are some constants, the values of which can be determined whenever the WFs are chosen by using the two steps of the procedure stated above. Equations (5), (9), (12), (13), (17) and (18) constitute the equations of the mth order

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approximate theory. These equations when accompanied by lateral boundary conditions govern the dynamic behaviour of cylindrical shells. 3.3. We now comment on the edge boundary conditions to be used in conjunction with the proposed approximate theory. Let C be the edge boundary of the cylindrical shell (CS) and be perpendicular to the mid-surface of the shell. Furthermore, let n be the outer unit normal of C, which, in view of the yi co-ordinate system used for CS (see Figure 1), has the form n = (n1 , 0, n3 ). In the exact theory, as edge boundary conditions we specify one member of each of the pairs (ti , ui ), i = 1–3; (qn , u) at each point of C, where qn and ti are, respectively, the normal heat flux and traction components on C, and are related to heat flux and stress components by qn = nj qj and ti = nj tji . To obtain the edge boundary conditions for the mth order approximate theory, we apply the operator Lk (k = 0, . . . , m) to the exact edge boundary conditions, which, in view of the last equations, leads to the conditions that, one member of each of the triplets (nj tjik , uik ), i = 1–3, and (nj qjk , u k) is specified at each point of C' and for the range of k, k = 0, . . . , m, where C' designates the edge boundary line, which is the intersection of C with the mid-surface of the shell. From the last equations it follows that the number of edge boundary conditions for the mth order approximate theory is 4(m + 1) at each point of the edge boundary line C'.

4. DETERMINATION OF THE CONSTANTS APPEARING IN THE APPROXIMATE THEORY WHEN THE WF ARE ORTHOGONAL

In developing the approximate theory no restriction is imposed on the choice of weighting functions fj , but, the choice of orthogonal functions for the fj s simplifies the computation of the constants appearing in the approximate theory. In what follows, we first present the expressions for the constants when the WFs are orthogonal; then, we list the values of the constants when the WFs are Legendre polynomials. 4.1. s First we shall determine the constants cnk , c¯nk and c nk . To this end we apply the operator Lk (k = 0, . . . , m) to both sides of each of equations (7), and then solve the resulting equations for the aforementioned constants by using the orthogonality of the fj s. This gives cnk =

1 ck

g

1

−1

f'n fk dy¯2 ,

c¯nk =

1 ck

g

1

wfn fk dy¯2 ,

−1

c nk =

1 ck

g

1

13 wfn fk dy¯2 ,

(20)

−1

where ck =

g

1

fk2 dy¯2 .

−1

Next, we present the expressions defining the constants gi and g3 which appear in the CE of the FVs, equations (17) and (18). By taking into account the orthogonality of the WFs and using the procedure stated in the previous section, these expressions can be

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T 1 Values of cij j ZXXXXXXXXXCXXXXXXXXXV 0 1 2

i 0 1 2

0 1 0

0 0 3

0 0 0

T 2 Values of c¯ij

i

j ZXXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXXV 0 1 2

0 1 2

a 1/2R − 1/2R 2a R 2 + 1/4R (3R 2 − 1)a −3/2R

− 1/2R 2a) 3(R 2 + 1/2R 3a) 3(−R 3 2 − 1/4R (3R 2 − 1)a] 3[3/2R

2 + R /4(3R 2 − 1)a] 5[−3/2R 3 − (R /2)2(3R 2 − 1)a] 5[3/2R 2 2 (3R − 1) − R /8 5[−3/4R 2 − 9R 4)a] (−1 + 6R

obtained, after some manipulations, as

F2 J + Gck {f'k (1) − g fk (1)}, k = 0, 2, . . . , p G gk = g h(no summation on k), G 2 {f'k (1) − g−fk (1)}, k = 1, 3, . . . , p'G fck j g+ = f'p + 2 (1)/fp + 2 (1),

g− = f'p' + 2 (1)/fp' + 2 (1).

(21)

4.2. s The constants appearing in the approximate theory are cij c¯ij , c ij , gi and g3, the values of which depend on the WFs, and can be determined by using equations (20) and (21). From the study of these equations it may be observed that, in addition to their dependence on the WFs, gi and g3 depend also on the order of the theory, and the c ij on the shape of cylindrical shell. For the case in which the WFs are Legendre polynomials, the values of the constants T 3 Values of gi and g for various orders of the theory 3

Order (m) 0 1 2

Constant ZXXXXXXXXXXXXXXCXXXXXXXXXXXXXXV g0 g1 g2 g+ g− −3 −3 −10

— −15 −15

— — −35

3 3 10

1 6 6

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cij , c¯ij and (gi , g ) are presented in Tables 1, 2 and 3 respectively for the orders of the approximate theory up to m = 2. In Table 2 the definitions 3

R = R/h,

a = ln {(R + 1)/(R − 1)}

(22)

are used. A dash in Table 3 designates that the corresponding constants does not exist in the approximate theory. The values of the c ij s are not given in the tables. As stated previously, the c ij s depend on the shapes of the cylindrical shells. They are all zero for circular shells.

5. DISCUSSION

The mth order approximate theory developed in this study has two types of field variables; namely, WVs and FVs. Through the use of FVs, the lateral boundary conditions of the shell can be accounted for correctly, and thus it may be expected that the geometric dispersive characteristics of thermoelastic shells may be well predicted by the proposed approximate theory. This anticipation has been verified for elastic plates and cylindrical shells in references [1] and [8] by comparing exact and approximate dispersion curves. Since the present proposed theory contains the theories developed in references [1] and [8] as its special cases, it is believed that it is also capable of predicting satisfactorily the behaviour of thermoelastic shells. The belief is indeed verified in the second part of the study for hollow thermoelastic cylindrical shells. The proposed theory has the following general features. (a) It incorporates the inhomogeneity in axial and circumferential directions of the shell. (b) The thickness of the shell may vary in axial and circumferential directions. (c) The shape of the cylindrical shell is arbitrary; i.e., the radius of curvature may vary in the circumferential direction. (d) The theory governs all of the deformation modes; i.e., it can be used for analyses involving flexural, axial, torsional, radial, etc., deformations of the shells. (e) The order of the approximate theory is arbitrary. By increasing the order, one can refine the theory and increase the thickness of the shell as much as one wishes, but at the expense of complicating the approximate theory.

REFERENCES 1. Y. M 1980 International Journal of Solids and Structures 16, 1155–1168. A new approach for developing dynamic theories for structural elements, part 1: application to thermoelastic plates. 2. E. H 1966 American Institute of Aeronautics and Astronautics 4, 2234–2236. Thermal excitations of thin elastic shells. 3. E. J. MQ and M. A. B 1970 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 37, 661–670. Dynamic thermoelastic response of cylindrical shells. 4. Y. T and T. F 1981 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 48, 113–118. Some considerations on thermal shock problems in a plate. 5. J. F. D 1988 Journal of Thermal Stresses 11, 175–185. Spectral analysis of coupled thermoelastic waves. 6. H. R and E. G. L 1988 Journal of Thermal Stresses 11, 77–91. Thermal vibrations of thin cylindrical shells.

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7. Y. M, G. B and H. D. MN 1980 International Journal of Solids and Structures 16, 1169–1186. A new approach for developing dynamic theories for structural elements, part 2: application to thermoelastic layered composites. 8. G. A. B and Y. M 1989 Journal of Sound and Vibration 130, 69–77. A refined dynamic theory for viscoelastic cylindrical shells and cylindrical laminated composites, part 2; an application.