Buckling and dynamic analysis of piezothermoelastic composite cylindrical shell

Composite Structures 59 (2003) 45–60 www.elsevier.com/locate/compstruct

Buckling and dynamic analysis of piezothermoelastic composite cylindrical shell N. Ganesan *, Ravikiran Kadoli Machine Dynamics Laboratory, Department of Applied Mechanics, Indian Institute of Technology, Chennai 600 036, India

Abstract Piezoelectric composite cylindrical shells operating in a steady state axisymmetric temperature are analyzed using a semi-analytical finite element method. Numerical studies are conducted on a clamped–clamped composite cylindrical shell for three typical length to radius ratios. The static thermal buckling analysis is carried out for a single layered composite cylindrical shell with different fiber orientation and hence study its influence on thermal buckling temperature. The influence of axisymmetric temperature on the natural frequencies and active damping ratio of the piezoelectric cylindrical shell is also examined. Two configurations of composite laminates: symmetric and antisymmetric with varying the number of plies is also considered to examine the effect on thermal buckling temperature. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Semi-analytical finite element; Axisymmetric temperature; Thermal buckling; Active damping ratio

1. Introduction In the recent few years, there is an accelerated effort and notable contributions on the study of electro–thermomechanical coupling behavior in smart structures. High operating temperatures are usually encountered in space shuttles, supersonic airplanes, rockets and missiles, as well as major load carrying elements of engines, etc. Many of the structural components have shell like configuration with most of the features close to that of circular cylinder. Composite materials are used extensively and they are subjected to thermal loading due to aerodynamic heating. Hence, thermal buckling becomes a major concern in structural component design. Hoff [1] discusses in detail the thermal buckling of thin circular cylindrical shells and columns under three categories namely, small uniform temperature raise, large temperature variation together with thermal expansion of structural elements, and time effects of creep. Hoff’s paper also presents a survey of the solutions to problems falling under these headings. Abir and Nardo [2] considers the problem of thermal buckling in thin circular cylindrical shells when there exits temperature gradients *

Corresponding author. Tel.: +91-44-2351365; fax: +91-442350509. E-mail address: [email protected] (N. Ganesan).

in the circumferential direction. Such situations are important in the structural design of high-speed airplanes and missiles. Lu and Chang [3] have worked on the thermal buckling of truncated conical shells when temperature varies along the generator and secondly temperature changes in the two-principle directions. Evaluation of the critical temperature and its relation to geometric parameters is made. Earl Thornton [4] has compiled an exhaustive review of the research on thermal buckling of plates and shells. The linear thermal buckling behavior of laminated composite shell under thermal load and the effect of various other parameters on the critical buckling temperature was studied by Thangaratnam et al. [5]. According to studies of Tzou and Ye [6] on piezoelectric laminated steel beams, and Sunar and Rao [7] on piezoelectric bimorph fingers, temperature variations generally have far reaching influences on the sensing and controlling capabilities of the sensor and actuator materials used for vibration control and this factor should be taken into account for precision control and actuation. Recent research on the behavior piezothermoelasticity aims at modeling and determining solutions accounting the interaction of the three fields. Analytical three-dimensional solutions of the fully coupled thermoelectroelastic response of multilayered hybrid composite cylindrical shells were reported by Xu and Noor [8], the Frobenius method and

0263-8223/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 2 3 0 - 1

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a sublayer method are used to obtain the response and sensitivity coefficients. Chen and Shen [9] have reported their analytical study to explain the exact mechanical and electric behavior of a laminated piezothermoelastic circular cylindrical shell subjected to axisymmetric thermal or mechanical loading. Gu et al. [10] developed a higher order temperature theory to determine the temperature distribution in laminated structures, this temperature distribution is used in the coupled thermopiezoelectric-mechanical theory for accurate analysis smart composite structures. Lee and Saravanos [11] have considered the piezoelectric coupling that arises from the pyroelectric and thermal expansion coefficients and developed a formulation capable of accurately predicting the mechanical, electrical, and thermal behavior of thin and moderately thick piezoelectric composite shells. The above literature deals with static thermal behavior of piezoelectric shells. Tzou and Bao [12] have derived generic nonlinear piezothermoelastic shell equations. Tzou and Zhou [13] used these equations to study the nonlinear behavior, von Karman type, of the circular plate. Very recently a comprehensive review on the status of piezothermoelasticity in particular reference to smart composite structures is presented by Tauchert et al. [14]. To the authors knowledge the dynamics of piezothermoelectric behavior of composite shells has not been studied so far in literature. In this paper a nonlinear piezothermoelastic characteristics of piezoelectric composite shells is attempted taking into account the initial stresses due to steady state axisymmetric temperature. It is assumed that linear vibrations occur in the vicinity of non-linear static equilibrium state of the system similar to Tzou and Zhou [13]. Semi-analytical finite element formulation is developed for the analysis of piezoelectric composite shells of revolution under an axisymmetric temperature distribution. First order shear deformation theory is used to formulate the structural finite element based on Ramalingeswara Rao [15]. A quadratic line element with three nodes is used to model the general composite shells of revolution. The active control continuum comprise of a thin layer of the spatially distributed piezoceramic convolving sensors and actuators pioneered by Tzou et al. [16], and also implemented by Saravanan et al. [17]. By considering the initial stresses in the smart composite structure due to axisymmetric thermal loading, the system can be characterized with the geometric stiffness matrix and enables a true study of the dynamic characteristics of piezothermoelastic continua. The two linear piezoelectric constitutive equations are solved assuming negligible induced potential in case of direct piezoelectric effect and negligible mechanical strain in the case of converse piezoelectric effect. This yields the active damping matrix of the system. Numerical studies related to thermal buckling and dynamic analyses are carried out on piezothermoelastic composite cylindrical

shells having three different l=r ratio for clamped– clamped boundary condition. The influence of fiber orientation on the thermal buckling temperature is examined. The study is extended for two configurations of composite laminates to determine the effect of increasing the number of plies on the critical thermal buckling temperature.

2. The physical model An interaction involving the mechanical, electrical and thermal field constitutes the piezothermoelastic phenomenon. Fig. 1 shows the schematics of the piezothermoelastic continua, which comprise of a composite cylindrical shell of length l, mid-surface radius r, and thickness h. The outer and inner surface of the shell is suitably bonded with piezoceramic material. These distributed sensors and actuators are of the convolving type, cos kh, shaped. These along with a negative velocity feedback controller constitute the active controlled continuum. The system comprising of the piezoceramic bonded composite shell is exposed to steady state heat source, q. Quantities qsi and qs0 represents a steady heat flux into and out of the system. The system operates under steady state temperature and temperature variation through the continua is axisymmetric. The continua are modeled in tri-orthogonal curvilinear coordinate system (s; h; z) as shown in Fig. 2. The three fundamental equations of the piezothermoelastic constitutive relations are the stress equation (a Duhamel–Neumann equation), the electric displacement equation and entropy density equation, Tzou and Ye [6], and are expressed respectively as follows rij ¼ cijkl ðekl  e0kl Þ  ekij Ek  kij T

ð1Þ

Di ¼ eikl ðekl  e0kl Þ þ nik Ek þ pi T

ð2Þ

g ¼ kkl ðekl  e0kl Þ þ pk Ek þ

qcv T0

ð3Þ

Fig. 1. Pizoelectric cylindrical shell continuum and Steady state heat source.

N. Ganesan, R. Kadoli / Composite Structures 59 (2003) 45–60

Fig. 2. Tri-orthogonal curvilinear coordinate system for shells of revolution.

where rij is the stress tensor, cijkl is the elastic coefficients at the constant electric field and temperature, ekl is the strain tensor, e0kl is the initial strain vector, eijk is the piezoelectric coefficients at a constant temperature, Ek is the electric field, kij is the temperature-stress coefficient, T is the temperature rise from stress free reference temperature T0 . Di is the electric displacement component, nik is the dielectric coefficient at the constant elastic stress and temperature, pi is the pyroelectric constant, g is the entropy per unit volume, cv is the specific heat at constant volume, q is the density of the material.

3. Semi-analytical finite element for piezothermoelasticity 3.1. Structural stiffness matrix In the first order shear deformation theory, the displacement field of the structural continuum are expressed in curvilinear coordinate system as the total displacement comprising of the sum of the mid-surface displacements u0 , v0 , w0 along the s, h, and z direction and rotations of the normal to the mid-surface ws ; wh along s and h axes respectively.

47

8 9 2 3 u0 > cos mh 0 0 0 0 > > > > > > > 6 7 > > 0 sin mh 0 0 0 7 > 1 6 < v0 > = X 6 7 6 0 0 cos mh 0 0 7 w0 ¼ 6 7 > > > > m¼0 6 7 > 0 0 cos mh 0 5 ws > > > 4 0 > > > > : ; 0 0 0 0 sin mh wh 8 9 > > u0m > > > > > > > > v0m > > < =  w0m ð7Þ > > > > > > wsm > > > > > > : ; whm

m stands for the mth circumferential harmonic. The strain–displacement relations for a doubly curved shell of revolution based on FSDT, adopted from Rao [15], are as follows ess ¼ 1=A1 ðe0ss þ zj1s Þ ehh ¼

1=A2 ðe0hh

þ

zj1h Þ

chz ¼ 1=A2 c0hz

ð8a; bÞ

1=A1 c0sz

ð8c; dÞ

csz ¼

csh ¼ 1=A1 1=A2 ðc0sh þ zj1sh Þ

ð8e; fÞ

where ð1=A1 Þ ¼ ð1=ð1 þ z=R/ ÞÞ and ð1=A2 Þ ¼ ð1=ð1 þ z=Rh ÞÞin the above equations R/ and Rh are the principle radii of curvature of the shell as illustrated in Fig. 2. ess , ehh , chz , csz , csh are the total strains which comprise of e0ss , e0hh , c0hz , c0sz , c0sh , the normal strains and the shear strains referred to mid-surface and j1s , j1h , j1sh , the change in curvature. The total strain energy in the composite continuum is given by U ¼ U1 þ U2 where U1 is the strain energy due to vibratory stresses and U2 is the strain energy contribution from the initial stresses developed due to steady state temperature. Strain energy U1 is given by Z 1 U1 ¼ fess rss þ ehh rhh þ chz shz þ csz ssz þ csh ssh g dV 2 V Z 1 T fg frg dA ð9Þ ¼ 2 A

uðs; h; z; tÞ ¼ u0 ðs; h; tÞ þ zws ðs; h; zÞ

ð4Þ

where frg and feg are the generalized stress and strain vectors respectively and dA is the infinitesimal area element on the shell mid-surface. The foregoing vectors are defined as

vðs; h; z; tÞ ¼ v0 ðs; h; tÞ þ zwh ðs; h; zÞ

ð5Þ

fg ¼ fe0ss e0hh c0sh j1s j1h j1sh c0sz c0hz g

ð10Þ

frgT ¼ fNss Nhh Nsh Mss Mhh Msh Qs Qh g

ð11Þ

wðs; h; z; tÞ ¼ w0 ðs; h; tÞ

ð6Þ

T

The generalized stress vector can be expressed as frg ¼ ½Dfg

The generalized displacement field is assumed to depend in the circumferential direction hence these quantities can be expanded by a Fourier series in the h direction, expressed as

ð12Þ

1 T U1 ¼ fde g ½ke fde g ð13Þ 2 where ½ke  is the element stiffness matrix corresponding to the mth harmonic and is computed as follows

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N. Ganesan, R. Kadoli / Composite Structures 59 (2003) 45–60

Q12 ¼ c2 s2 ðQ11 þ Q22  4Q66 Þ þ ðc4 þ s4 ÞQ12 Q13 ¼ c2 Q13 þ s2 Q23 Q16 ¼ c3 sðQ11  Q12  2Q66 Þ þ cs3 ðQ12  Q22  2Q66 Þ Q22 ¼ s4 Q11 þ c4 Q22 þ 2c2 s2 ðQ12 þ 2Q66 Þ Q23 ¼ s2 Q13 þ c2 Q23 Q26 ¼ cs3 ðQ11  Q12  2Q66 Þ þ c3 sðQ12  Q22  2Q66 Þ Q33 ¼ Q33 Q36 ¼ csðQ13  Q23 Þ Fig. 3. Schematics of the discretization of the cylindrical shell using 3 noded quadratic line element.

½ke  ¼

Z

T



½B  ½D½B r ds dh

ð14Þ

A

Due to orthogonality principle the stiffness matrix become decoupled for each circumferential harmonic. A three noded isoparametric element is used in the axial, s-direction and Fig. 3 depicts the dicretization of the composite cylindrical shell using this element. Each node has five degrees of freedom. The displacement parameters associated with the element are fde g ¼ fu1 ; v1 ; w1; ws1 ; wh1 ; . . . ; ws3 ; wh3 g

ðb2  bÞ ; 2

ðb2 þ bÞ 2 ð16a–cÞ The constitutive matrix ½D in Eq. (14) is for a laminate consists of various integrated shell stiffness Aij , B ij , Dij , F ij , H ij ; . . . Various integrated shell stiffness are defined as Z þh=2 ðAij ; Bij ; Dij ; Fij ; Hij ; . . .Þ ¼ Qij ð1; z; z2 ; z3 ; z4 ; . . .Þ dz N2 ¼ ð1  b2 Þ and

Q45 ¼ csðQ55  Q44 Þ Q55 ¼ s2 Q44 þ c2 Q55 Q66 ¼ c2 s2 ðQ11 þ Q22  2Q12  2Q66 Þ þ ðc4 þ s4 ÞQ66 c ¼ cos # and s ¼ sin #, in which # is the fiber orientation angle with reference to the s-axis. Qij are expressed in terms of elastic moduli in principal material coordinate directions for an orthotropic material and are given by Jones [18]. 3.2. Geometric stiffness matrix evaluation

ð15Þ

the subscripts 1, 2, and 3 stand for the three nodes of the element. The shape functions Ni in terms of the isoparametric axial coordinate b ¼ ðs=lÞ are given by N1 ¼

Q44 ¼ c2 Q44 þ s2 Q55

N3 ¼

h=2

ð17Þ in which Qij are the reduced stiffness coefficients given by the constitutive relationship, 38 9 2 9 8 > > rss > ess > > > > 0 0 Q16 7> > > > > 6 Q11 Q12 > > > > > > > > 7 6 > Q22 0 0 Q26 7> = < ehh > = 6 < rhh > 7 6 ¼6 shz Q44 Q45 0 7 chz > 7> > >c > > 6 >s > > > 6 Q55 0 7 sz > > > > sz > > > > > 5> > > ; 4 : > ssh ; : csh > Sym: Q 66

Neglecting the strain energy due to initial transverse shear stresses, strain energy due to initial stresses U2 as given in Rao [15] are, may be written as Z n o 1 2 2 2 ðeiss Þ r ss þ ðeihh Þ r hh þ 2ðcish Þ s sh dV U2 ¼ ð19Þ 2 V in which r ss , r hh , and s sh are the initial stresses and the initial strains, eiss , eihh and cish are given as follows   i 1 ow ui i   ess ¼ ð20Þ  os R/ 1 þ z=R/   1 1 owi vi  ð21Þ eihh ¼   sin / r oh r 1 þ z=R/ cish ¼ eiss eihh

ð22Þ

Eq. (19) can be put in the following form 1 T ð23Þ U2 ¼ fdie g ½kre fdie g 2 The matrix ½kre  in the foregoing equation is the element geometric stiffness matrix Z ½kre  ¼ ½Bi T fr g dV ð24Þ V

fdie g represents the elemental displacement vector due to initial strain.

ð18Þ T

Qij ¼ ½T ½Q½T is the transformed elasticity matrix in global coordinates, Q11 ¼ c4 Q11 þ s4 Q22 þ 2c2 s2 ðQ12 þ 2Q66 Þ

3.3. Thermal load and initial stress evaluation Under thermal environment the expression for the total potential for an element is as follows

N. Ganesan, R. Kadoli / Composite Structures 59 (2003) 45–60

1 P¼ 2

Z

T

fg ½Dfg dV 

V

Z

T

fg fkgT dV

ð25Þ

V

The temperature is assumed to be constant over the thickness and the temperature field for the composite structural and piezoceramic continua is T ðs; h; z; tÞ ¼ T ðs; h; tÞ Steady state temperature is assumed to be dependent in the circumferential direction, and using Fourier series the temperature in the circumferential direction is 1 X T ¼ Tm cos mh; ð26Þ m¼0

Minimization of P with respect to the displacement vector fde g leads to a standard equations ½ke fde g  fFth e g ¼ 0

ð27Þ

½ke  is the structural element stiffness matrix same as Eq. (14). The thermal load evaluation involves: Z T fFth g ¼ ½ fkgfTðzÞg dV ð28Þ where fkg is the temperature stress coefficients computed for the laminate as nlay Z hk X ½Qij ðkÞ f agðkÞ ð1; zÞ dA ð29Þ fkg ¼ k¼1

49

fe0 g, represents the strain developed in the structural continua due thermal load, ½D is matrix comprising of various integrated shell stiffness in the transformed coordinate, and fkg represents the temperature stress coefficients, and is similar to the one in Eq. (29). The stress resultants are found for each finite element and are used in Eq. (24) to compute the element geometric stiffness matrix. 3.4. Mass matrix evaluation The mass matrix is obtained from the kinetic energy of the shell continuum, the kinetic energy is Z Z q q T KE ¼ ðu_ 2 þ v_ 2 þ w_ 2 Þ dV ¼ fd_ g fd_ g dV ð31Þ 2 V 2 V Using Eq. (16) the kinetic energy will be 1 T KE ¼ fde g ½me fde g 2 where ½me  is the element mass matrix given by Z me ¼ q NT N dV

ð32Þ

V

3.5. Active control under the steady-state temperature field

hk1

ðkÞ

½Qij  represents the generally orthotropic lamina properties of the kth layer with proper coordinate transformation and comprises of the stiffness coefficients Aij and Bij . The coefficient of linear expansion for the kth layer in the shell coordinate system is k

f ag ¼ fas ah ash 00000g

T

where as ¼ a1 cos2 # þ a2 sin2 # as ¼ a1 sin2 # þ a2 cos2 # as/ ¼ ða1  a2 Þ cos # sin # # is the lamina orientation angle, T ðzÞ is the temperature ðkÞ rise from stress free temperature T0 , fag is the vector of coefficient of linear thermal expansion in the kth layer, a1 and a2 are the coefficients of thermal expansion in the directions parallel and perpendicular to the lamina or (coordinate axis for kth lamina). Using Eq. (28) the thermal load is computed for a steady state axisymmetric temperature, above the stress free temperature T0 . Solving Eq. (27) for the whole composite structural continua gives the displacement field developed due to the operation of the composite cylindrical shell in thermal environment. Using the element displacement vector the stresses and moment resultants are evaluated as follows frg ¼ ½Dfo g  fkgT

ð30Þ

The two piezothermoelastic constitutive Eqs. (1) and (2) viz. the electric displacement equation and Duhamel–Numann equation respectively will be used to obtain the element active damping matrix. The piezoceramic sensor layer is very thin of the order of 0.02 mm, hence the poling direction can be taken along the z-direction only. The potential u in the piezoelectric layer is assumed to vary linearly over the thickness, hence   u T fEg ¼ 0 0  ð33Þ tp where tp is the thickness of the piezoceramic sensor layer. Making use of the direct piezoelectric effect and assuming the external charge applied is zero one can compute the sensor output, based on the theory developed for a distributed piezoelectric sensing by Tzou [19,20]. 0 ¼ eikl ekl þ nik Ek þ pi T

ð34Þ

Charge generated by the sensor device is proportional to the strain spatially integrated over the area they occupy, and considering an open-circuit condition, the sensor output from Eq. (34) becomes Z As

½nfEg dA ¼

Z

½efg dA þ As

Z

fpgT dA A

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N. Ganesan, R. Kadoli / Composite Structures 59 (2003) 45–60

  Z   Z 1 ess us ¼ ts ½e31 e32  dA þ ts fpgT dA ehh n33 Aelec Aelec A ð35Þ Note that Eq. (35) gives us the averaged charge and Aelec is the area of the electrode. The constitutive relation, Eq. (1), that defines the converse piezoelectric effect is used to determine the actuating force generated by distributed actuators. rij ¼ cijkl ekl  ekij Ek  kij T Assuming the mechanical induced strain to be negligible in the above equation and initial stresses due to axisymmetric temperature are accounted in geometric stiffness matrix, the stress component in the actuator layer will be rij ¼ ekij Ek

ð36Þ

From the principle of virtual work the actuating force is computed as outlined below Z T T T fdg frg ¼  fdg ½e fEg dV ð37Þ V

From the above equation, the force in the actuator can be deduced as   ua fFg ¼ ½kauu  ð38Þ ta where ua is the actuator potential and ta is the thickness of actuator. The controller is a negative velocity feedback defined as ua ¼ GF u_ s

ð39Þ

where u_ s is the time derivative of us and GF is the feedback factor, defined as the ratio of the charge (or voltage) supplied to the actuator charge (or voltage) generated by the sensor. Substituting Eq. (35) after taking the time derivative into Eq. (39) the actuator voltage will be  Z  Z 1 _ _ ua ¼ GF ts ½e½B dAfdg þ ts fpgT dA n33 Aelec Aelec A ð40Þ Since the temperature is assumed to be steady, the time derivative of temperature is zero, accordingly substituting Eq. (40) in Eq. (38) the actuator force is given as   Z  1 fFg ¼ ½kauu   GF ts ½e½B dAfd_ g n33 Aelec Aelec   fFg ¼ ½kauu  ð  GF Þ½ksuu  ð41Þ The above equation represents the active damping matrix ½cae  for a element.

4. Prebuckling analysis and equation of motion The prebuckling solution is obtained for the specified temperature distribution by keeping an arbitrary magnitude of the same. The following classical equation is used to perform static thermal buckling analysis ðK þ vKr Þxb ¼ 0

ð42Þ

In the above equation K is the structural stiffness matrix, Kr is the geometric stiffness matrix, contribution from the thermal load due to steady state axisymmetric temperature. v are the buckling eigenvalues which multiplies the applied temperature to give the critical buckling temperature and fxb g is the mode shape. Using Lagrange’s equation, the equation of motion for an element is me €de þ cae d_ e þ ðke þ kre Þde ¼ 0 and the global equation of motion will take the form ½Mf€dg þ ½Ca fd_ g þ ½K þ Kr fdg ¼ 0

ð43Þ

5. Validating the formulation 5.1. Validation of critical buckling temperature Thangarathnam [21] have carried out thermal buckling analysis of composite cylindrical shells under the influence of mechanical and thermal loads using semiloof finite element. Studies of the results on buckling of symmetric cross ply composite cylindrical shell under uniform temperature with simply supported boundary condition were presented. The results are simulated using the present formulation for the same composite cylindrical shell. The laminate properties used as quoted in Thangaratnam are as follows: E11 =E22 ¼ 10;

G12 =E22 ¼ 0:5;

l12 ¼ 0:25; 5

a22 =a11 ¼ 2:0 and a11 ¼ 0:1  10 : For a typical length to radius ratio ðl=rÞ ¼ 0:5, and for different ratio of radius to thickness ðr=hÞ, the critical buckling temperature is computed using the present code. The results obtained are compared with those reported by Thangaratnam and are presented in Table 1. Table 1 Validation of critical buckling temperature for symmetric cross-ply (90/0/90) cylindrical shell r=h 200 300 400 500 a

Critical temperature (°C) Reference [21]

Results from present study

1304.298 (11)a 912.434 (13) 659.610 (18) 514.745 (19)

1258.4 (11), 1250 (12), 1268.4 (13) 851.3 (13), 835.8 (14), 836.6 (15) 632.2 (16), 631 (17), 637.8 (18) 507.3 (18), 507 (19), 511.74 (20)

Circumferential mode.

N. Ganesan, R. Kadoli / Composite Structures 59 (2003) 45–60

51

(c) The influence of number of plies and fiber orientation, for laminate configuration like antisymmetric and symmetric, on the critical buckling temperature. Numerical studies are presented for two fiber angle 15° and 60° and for l=r ¼ 0:54 and l=r ¼ 2:08. (d) The variation of the first axial mode natural frequency associated with different circumferential modes, with respect to temperature limited to critical buckling temperature. Two piezoelectric composite shells of l=r ¼ 0:54 and 2.08 are considered for the study. (e) The variation of the active damping ratio in the first axial mode associated with different circumferential modes, with respect to axisymmetric temperature limited to critical buckling temperature. Two piezoelectric composite shells of l=r ¼ 0:54 and 2.08 are considered for the study.

The trend in the results of critical buckling temperature obtained from the present approach tally well with that of Thangaratnam. It is also seen that the circumferential mode at which thermal buckling occurs are in close agreement.

6. Numerical simulations––results and discussion Numerical studies are conducted on piezoelectric composite cylindrical shells dealing with: (a) Static thermal buckling analysis, to determine the critical buckling temperature of the first axial mode associated different circumferential modes. Numerical studies are also made to evaluate the influence of fiber angle on the thermal buckling temperature. Typical composite shells of l=r ¼ 0:54, 1.048 and 2.08 and each having r=h ¼ 292 are considered in the study. (b) Numerical procedure to determine the best fiber angle for composite ply which will possess high critical thermal buckling temperature, since it is possible to tailor the manufacture of the composite materials.

The geometric details of the cylindrical shell are given in Table 2. The properties of piezoceramic material (PZT4) and composite shell material, HS-Graphite/Epoxy are listed in Table 3. It is assumed that the material properties do not vary with temperature.

Table 2 Details of composite cylindrical shell and boundary conditions Cylindrical shell

Length l (m)

Radius r (m)

l=r ratio

Thickness of cylindrical shell (m)

Boundary condition

1 2 3

0.4572 0.914 1.8288

0.876 0.876 0.876

0.54 1.04 2.08

0.003 0.003 0.003

Clamped–clamped Clamped–clamped Clamped–clamped

Table 3 Material properties [18,21–23] HS-graphite/epoxy

Piezoceramic (PZT4)

Young’s modulus (GPa)

E11 E22 E33

180.8 10.4 10.4

81.3 81.3 64.5

Shear modulus (GPa)

G12 G31 G23

7.234 7.234 7.234

30.6 25.6 25.6

Poisson’s ratio

m12 m13 m23

0.28 0.28 0.28

0.329 0.432 0.432

Density (kg/m3 )

q

1389.23

7600

Piezoelectric properties (C/m2 )

E31 e32

– –

5.20 )5.20

Dielectric constant (C2 /(Nm2 ))

n33

Coefficient of thermal expansion (/°C)

a11 a22

11:34  106 36:9  106

1:2  106

Thermal stress coefficient (N/m2 °C)

k

1:08  106

1:03  105

Pyroelectric constant (C/m2 °C)

P

–

0:25  104

Ambient temperature (°C)

T1

28

28

11505:0  1012

52

N. Ganesan, R. Kadoli / Composite Structures 59 (2003) 45–60

6.1. Thermal buckling of composite cylindrical shell under steady-state axisymmetric temperature The variation of first axial mode critical buckling temperature of composite cylindrical shells with respect to circumferential harmonic are illustrated in Fig. 4(a) and (b). Composite shells with fiber angles 0°, 15°, 30°, 45°, 60°, 75°, and 90° are considered in the study. From Fig. 4(a) and (b), it is to be noted that the critical buckling temperature is highly dependent on the fiber angle. The critical thermal buckling temperature for composite shells with fiber angle 0°, 15° and 30° does not vary appreciably form harmonic to harmonic. The critical thermal buckling temperature for composite shell with fiber angle 45° and 60° increases as the number of harmonics increase. For fiber angle 75°, 90° the thermal buckling temperature decreases as the number of harmonics increase until the temperature attains lowest value at certain harmonic number, and then increases as the number of harmonics increase.

Fig. 4. Plots of critical buckling temperature for the first axial mode associated with different circumferential modes m. Clamped–clamped composite cylindrical shell: (a) l=r ¼ 0:54 and (b) l=r ¼ 1:048.

Table 4 Magnitude of critical buckling temperature for the lowest first axial mode Fiber angle (degree)

Critical buckling temperature (°C) l=r ¼ 0:54

l=r ¼ 1:048

l=r ¼ 2:08

0 15 30 45 60 75 90

35 (m ¼ 14) 22 (m ¼ 12) 32 (m ¼ 1) 95 (m ¼ 1) 235 (m ¼ 1) 275 (m ¼ 17) 181 (m ¼ 14)

31 (m ¼ 14) 19 (m ¼ 13) 31 (m ¼ 11) 102 (m ¼ 1) 298 (m ¼ 1) 272 (m ¼ 13) 175 (m ¼ 14)

29 (m ¼ 14) 18 (m ¼ 13) 30 (m ¼ 10) 122 (m ¼ 1) 285 (m ¼ 11) 221 (m ¼ 11) 172 (m ¼ 13)

Number in bracket corresponds to the circumferential mode.

Irrespective of the fiber angle the axisymmetric mode has the highest thermal buckling temperature considering the number of harmonics presented here. In order to obtain an insight about the influence of the fiber angle on the thermal buckling temperature, the magnitude of the critical buckling temperature corresponding to the lowest first axial mode for different fiber orientation is considered and the same is listed in Table 4. It is observed from the Table 4, as l=r increases the critical thermal buckling temperature decreases with the exception for l=r ¼ 2:08 for fiber angle 45° and 60°. Composite shells with fiber angle 45°, the critical buckling temperature increases as the l=r increases, whereas for composite shell with fiber angle 60° the critical buckling temperature for l=r ¼ 1.048 is greater than that for l=r ¼ 0:54 and 2.08. Table 4 also reveals that composite shell with fiber angle 15° buckles at much lower temperature. Composite shells with fiber orientations above 45° up to 90° can be operated at much higher temperature compared to composite shells with fiber angle 0° above and up to less than 45°. The buckling behavior of composite shell and its dependence on fiber orientations can be explained by evaluating the magnitudes of the stress resultants and moment resultants at different points along the length of the shell. Fig. 5(a)–(c) illustrates the variation of stress resultants: Ns , Nh , and Nsh , and Fig. 6(a)–(c) the moment resultants: Ms , Mh , and Msh , along the length of the composite shell for an arbitrary steady state axisymmetric temperature of 30 °C above the stress free temperature. Plots illustrated are for a typical composite shell with l=r ¼ 1:048 for different fiber orientation. The trend in the results for composite cylindrical shell of l=r ¼ 0:54 and 2.08 are similar. From the plots in Fig. 6(a)–(c), in general it is seen that the values of moment resultants are considerably lower in magnitude when compared to the magnitude of stress resultants, Fig. 5(a)–(c). In general, for composite shells with different l=r ratios considered in the study the magnitude of membrane stress resultants Ns are more or less constant along the length of the shell. In contrast the

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Fig. 6. Variation of moment resultants along the merdian (s-axis) of the shell. l=r ¼ 1:048; clamped–clamped boundary conditons; 30 °C steady state axisymmetric temperature. Fig. 5. Variation of stress resultants along the merdian (s-axis) of the shell. l=r ¼ 1:048; clamped–clamped boundary conditions; 30 °C steady state axisymmetric temperature.

behavior of Nh is different from that of Ns and it shows considerable magnitude near the clamped edges of the shell and over the inner regions along the length of shell the magnitudes are very small. Hence it is clear that the influence of Ns will be more on the thermal buckling behavior compared to that of Nh . From Fig. 5(c), the variation of Nsh is highly dependent on the fiber angle laid for the composite shell. Also in the case of composite shell with fiber angle 60°, 75° and 90° the variations of Nsh follows a similar pattern of Nh , the magnitudes of both are appreciably great over the small length from the fixed end of the shell. The difference is

that Ns shows up itself with small magnitude along the length of shell whereas Nh is zero, refer Fig. 5(b) and (c). The membrane stress resultants Ns for composite shell with fiber angle 15°, 30°, and 0° are considerably large in magnitude along the length of the shell in that order. The membrane stress resultants are comparatively low for composite shell with fiber angle 60°, 75°, and 90°. Also note from Fig. 5(c), composite shell with fiber angle 30° and 15° develop considerable magnitude of stress resultants Nsh all along the length of shell. For composite shells with fiber angle 45°, the membrane stress resultants are reasonably high as in Fig. 5(a) when compared to composite shells with fiber angle 60°, 75°, and 90°. Tables 5–7 gives the magnitude of the stress resultants

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Table 5 Stress resultants Ns (N/m) for composite cylindrical shell Fiber angle (deg)

l=r ¼ 0:54 (0.015 m)

l=r ¼ 1:048 (0.030 m)

l=r ¼ 2:08 (0.061 m)

0.00 15.0 30.0 45.0 60.0 75.0 90.0

0.1882Eþ06 0.3536Eþ06 0.2673Eþ06 0.7429Eþ05 0.2778Eþ05 0.2172Eþ05 0.3009Eþ05

0.1866Eþ06 0.3459Eþ06 0.2532Eþ06 0.7053Eþ05 0.2609Eþ05 0.2090Eþ05 0.2999Eþ05

0.1861Eþ06 0.3434Eþ06 0.2499Eþ06 0.6985Eþ05 0.2444Eþ05 0.1944Eþ05 0.2979Eþ05

Table 6 Stress resultants Nh (N/m) for composite cylindrical shell Fiber angle (deg)

l=r ¼ 0:54 (0.015 m)

l=r ¼ 1:048 (0.030 m)

l=r ¼ 2:08 (0.061 m)

0.00 15.0 30.0 45.0 60.0 75.0 90.0

0.3649Eþ05 0.5379Eþ05 0.9173Eþ05 0.8130Eþ05 0.2025Eþ06 0.4143Eþ06 0.2306Eþ06

0.3441Eþ05 0.5008Eþ05 0.8202Eþ05 0.7272Eþ05 0.1826Eþ06 0.3730Eþ06 0.2070Eþ06

0.3126Eþ05 0.4534Eþ05 0.7400Eþ05 0.6624Eþ05 0.1616Eþ06 0.3091Eþ06 0.1672Eþ06

Table 7 Stress resultants Nsh (N/m) for composite cylindrical shell Fiber angle (deg)

l=r ¼ 0:54 (0.015 m)

l=r ¼ 1:048 (0.030 m)

l=r ¼ 2:08 (0.061 m)

0.00 15.0 30.0 45.0 60.0 75.0 90.0

0.0000Eþ00 0.9663Eþ05 0.1395Eþ06 0.5859Eþ05 0.8662Eþ05 0.1049Eþ06 0.6727Eþ04

0.0000Eþ00 0.9449Eþ05 0.1312Eþ06 0.5335Eþ05 0.7804Eþ05 0.9532Eþ05 0.6727Eþ04

0.0000Eþ00 0.9343Eþ05 0.1278Eþ06 0.5031Eþ05 0.6898Eþ05 0.8051Eþ05 0.6727Eþ04

prevailing at a Gaussian point on the shell very close to the clamped edge of the shell. From the Table 5 it can be observed that the composite shells with fiber angle 15°, 30° and 0° possess high magnitude of membrane stress resultants respectively, when compared to composite shell with fiber orientation 45°, 60°, 75°, and 90°. Since it is the membrane stress resultants, which governs the thermal buckling behavior, composite shells made with fiber angle 15°, 30° and 0° will buckle at very low operating temperatures, hence unsuitable for operation in thermal environments. One can tailor a composite shell with fiber angle 60°, 75° and 90° which develop very low order of axial compressive stresses due to uniform axisymmetric temperature, hence suitable for high temperature operating environments. General trend in the result revealed in Table 4 is that as the l=r increases the thermal buckling temperature decreases. On the contrary, a peculiar behavior of thermal buckling is noticed in case of composite shell for fiber angle 45° and 60°. As noted earlier, in case of

composite shell with fiber angle 45° the thermal buckling temperature increases as the l=r increases whereas in case of composite shell with fiber angle 60° the thermal buckling temperature is less for l=r ¼ 2:08 when compared to l=r ¼ 1:048. From this numerical study it is also seen that composite shells made with fiber angle 60° and 75° can withstand high operating temperature since the critical thermal buckling temperature is high. 6.2. Choice of fiber orientation from thermal buckling point of view: nomograph The above section provides enough evidence of the fiber angle governing the thermal buckling behavior of composite shells. It is known that the elastic modulus and coefficient of thermal expansion in the two principal material directions also influence the thermal buckling behavior of composite shells. Hence a general numerical study is taken up in order to ascertain the best orientation of the fiber for manufacture of composite shell, which possess different elastic modulus in the two principal direction as well as different thermal properties namely the coefficient of thermal expansion. Numerical study involves the evaluation of the membrane stress resultants for composite shells with different fiber angle for various ratios of material properties ðE11 =E22 Þ and in turn changing the ratio of coefficient of thermal expansion ða22 =a11 Þ. Typically the thermal expansion coefficient ratio ða22 =a11 Þ is varied from 2.0 to 10.0 and for each ratio of ða22 =a11 Þ the ratio ðE11 =E22 Þ is varied from 2.0 to 20.0. This ratio is so chosen as to cover most of the practically used composite materials like E-glass/ epoxy, Boron/Epoxy, HS-Graphite/Epoxy with the exception HM-Graphite/Epoxy, see Table 8. Typical results of the numerical study are presented in Fig. 7(a) and (b) for ða22 =a11 Þ ¼ 2:0 and 4.0, Fig. 7(a) and (b) for ða22 =a11 Þ ¼ 6:0 and 8.0 and Fig. 7(e) for ða22 =a11 Þ ¼ 10:0. From the graphs it is seen that the membrane stress resultants do not differ greatly for fiber angles 0°, 15°, 30° and 45°, irrespective of varying ðE11 =E22 Þ, for a given ða22 =a11 Þ ratio when compared to composite shell with fiber angle 60°, 75° and 90°. Thus for composite shell with fiber angle 0°, 15° and, 30° and 45° it can be inferred that the critical thermal buckling temperature may not differ greatly. As the ratio of ða22 =a11 Þ increases so does the membrane stress resultants irrespective of fiber angle and in turn they are dependent on the ratio ðE11 =E22 Þ, this is especially predominant in case of composite shell with fiber angle 60°, 75° and 90°. For a given ratio of ða22 =a11 Þ: (i) the membrane stress resultants are highest in magnitude for fiber angle 15° and 30° irrespective of the ratio ðE11 =E22 Þ and (ii) the stress resultants are higher in magnitude for composite shells with fiber angle 60° and 75° for ðE11 =E22 Þ ¼ 2:0 and 4.0 when compared to ðE11 =E22 Þ greater than 4.0.

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Table 8 Material properties of unidirectional composites from Thangarathnam [21] Composite material property 2

E11 (N/m ) E22 (N/m2 ) G12 (N/m2 ) E11 =E22 l12 a11 =°C a22 =°C a22 =a11

E-glass/epoxy 9

53.8  10 17.93  109 8.6  109 3.0 0.2.5 6.3  106 25.2  106 4.0

Boron/epoxy 9

209  10 17.6  109 5.17  109 11.85 0.20 3.96  106 20.16  106 5.0

It is interesting to note that for composite shell with fiber angle 0° the membrane stress resultants increases slightly as the ratio ða22 =a11 Þ increases from 2.0 to 20.0. Hence composite shell with 0° fiber angle will have higher thermal buckling temperature for higher ratio ða22 =a11 Þ, referring to Fig. 7(d) and (e) for ða22 =a11 Þ ¼ 8:0 or 10.0. In case of composite shell with fiber angle 90° the variation of the membrane stress resultants are highest for ðE11 =E22 Þ ¼ 2:0, 4.0 and 6.0 respectively in that order for a given ratio ða22 =a11 Þ. This trend is maintained as the ratio of ða22 =a11 Þ increases from 2.0 to 20.0. In general for a given ration of ða22 =a11 Þ the membrane stress resultants are highest for ðE11 =E22 Þ ¼ 2:0, reaching magnitude close to that of composite shell with fiber angle 15° and 30° as seen incase of ða22 =a11 Þ ¼ 10:0, Fig. 7(e). Care should be exercised when laying fiber angle 90° since for certain range of ða22 =a11 Þ and ðE11 =E22 Þ the thermal buckling behavior may be satisfactory for high temperature operation. From the above numerical study one can find, in general, that the composite shells with 15° and 30° fiber angles will have the lowest critical temperature. Composite shell with 60° and 75° fiber orientation will have high thermal buckling temperature when ðE11 =E22 Þ is greater than 8.0. By generating data such as the one illustrated here it is possible for the structural designer, from thermal buckling point of view, to chose a appropriate fiber angle for a given ratio of ða22 =a11 Þ and ðE11 =E22 Þ. 6.3. Influence of number of composite lamina on thermal buckling temperature Numerical studies are carried out to determine the influence of the number of plies recommended to manufacture composite shells based on the thermal buckling behavior. Different configurations of ply lay-ups considered for study are: symmetric laminates and antisymmetric laminates. As found in the Section 6.1 in the current study, 15° fiber orientation gives the lowest thermal buckling temperature and 60° fiber orientation can withstand high operating temperatures. Thus study has been carried out considering these two extreme fiber

HS-graphite/epoxy 9

181  10 10.34  109 7.2  109 17.5 0.28 11.34  106 36.9  106 3.25

HM-graphite/epoxy 206.8  109 5.2  109 3.1  109 39.7 0.25 )0.558  106 35.28  106 )63.0

angles for the lamina, from the point of view of thermal buckling behavior, for the laminates. Results are presented for cylindrical composite shell with l=r ¼ 2:08, total thickness of shell is 3.0 mm and for clamped– clamped boundary condition. 6.3.1. Symmetric laminated cylindrical shell Fig. 8 shows the schematics of a symmetric composite laminate. Table 9 shows the results for layered composite cylindrical shell made of symmetric angle ply. The critical thermal buckling temperature remains the same for the first axial mode associated with circumferential harmonics listed in table, even though the number of plies are increased, in the case of each lamina having fiber oriented at 15°. In case of laminated composite shell wherein each lamina is made of 60° fiber angle, the critical thermal buckling temperature depends on the number of layers used to manufacture the composite shell. 6.3.2. Antisymmetric laminated composite shell Fig. 9 shows the schematics of a antisymmetric angle ply laminate with six plies. The results of the numerical study are listed in Table 10. In case of the antisymmetric laminated composite shells, the critical buckling temperature increase as the number of plies for the composite shell laminate is increased. The increase is appreciable when the number of plies is increased from two to four. When the number of plies is increased beyond four, say 6 the critical buckling temperature does not increase much as seen from Table 10. Similar trend in the results were reported by Thangarathnam [21] using semi-loof element. 6.4. Variation of natural frequency with different axisymmetric temperature The temperature effect on the free vibration characteristic of the piezoelectric cylindrical shell in the vicinity of non-linear static equilibrium is studied. The variation of the square of the first axial mode natural frequency with respect to axisymmetric temperature for piezoelectric composite shell with l=r ¼ 0:54 and l=r ¼ 2:08 and for fiber angles 45°, 60°, 75° and 90° are displayed in

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Fig. 7. Variation of membrane stress resultants with respect to fiber angle, for different ratios of ðE11 =E22 Þ ¼ 2:0 to 20.0 and for ða22 =a11 Þ ¼ 2:0 and ða22 =a11 Þ ¼ 4:0. Variation of membrane stress resultants with respect to fiber angle, for different ratios of ðE11 =E22 Þ ¼ 2:0 to 20.0 and for ða22 =a11 Þ ¼ 6:0 and ða22 =a11 Þ ¼ 8:0. Variation of membrane stress resultants with respect to fiber angle, for different ratios of ðE11 =E22 Þ ¼ 2:0 to 20.0 and for ða22 =a11 Þ ¼ 10:0.

Figs. 10 and 11. The results are presented for first axial mode associated circumferential harmonic one, five, ten and for circumferential harmonic for which the first axial mode is the lowest in magnitude. The characteristics curves of the square of the natural frequency with respect to axisymmetric temperature for the first harmonic and its associated first axial mode frequency shows drastic reduction of natural frequency

as the temperature increases for fiber angle 45°, 60° and 75°, refer Fig. 10(a)–(c). These characteristics show high nonlinearity as the temperature approaches the critical temperature for which the frequency becomes zero. In case of composite shell with 90° fiber orientation, the frequency does not decrease as drastically as seen for other fiber angles, but does fall steeply as the temperatures nears the critical temperature, Fig. 10(d). The

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Fig. 8. Schematics of symmetric composite laminate with 5 plies. Fiber angle for each ply is indicated.

characteristic curves for the first axial mode associated with circumferential harmonics, m ¼ 5 and 10, see Fig. 10(a) and (b); m ¼ 5, 10, and 17, see Fig. 10(c); and m ¼ 5, 10 and 14, see Fig. 10(d), indicate that the fall in natural frequency of the shell is very less as the temperature increases. But the behavior becomes highly nonlinear when the temperature nears the critical temperature. At the critical thermal buckling temperaure, obviously the natural frequency of the system becomes zero. Similar trend in piezothermoelastic behavior has been reported by Tzou and Zhou [13] for a piezoelectric circular plate. For piezoelectric composite shell with l=r ¼ 2:08 and fiber orientation of 45°, 60°, 75° and 90° the characteristics curves of the square of the natural frequency with respect to axisymmetric temperature are displayed in Fig. 11. In general it is seen that the square of the nat-

Fig. 9. Schematics of antisymmetric composite laminate with 6 plies. Fiber angle for each ply is indicated.

ural frequency decreases slowly as the temperature is increased, and the nature of the curve shows a drastic fall in natural frequency as the temperature nears the critical temperature and the system natural frequency becomes zero. 6.5. Active damping studies of piezoelectric composite cylindrical shells with axisymmetric temperature distribution Typical results of the studies on the variation of active damping ratio with respect to axisymmetric temperature for a composite shell with l=r ¼ 0:54, fiber angle 45° and 60° and for clamped–clamped boundary

Table 9 Comparison of critical buckling temperature for layered composite shell: l=r ¼ 2:08; clamped–clamped boundary condition ðm; nÞ

Fiber angle 15°

60°

Number of lamina

(1,1) (5,1) (10,1) (15a ,1) a

3

5

7

3

5

7

27.6 27.00 26.23 24.92

27.27 27.06 26.46 25.38

27.27 27.07 26.49 25.47

738.3 760.93 495.68 411.0

750.9 811.62 484.97 416.93

754.11 790.15 487.91 418.66

Lowest circumferential mode corresponding to first axial mode.

Table 10 Comparison of critical buckling temperature for layered composite shell: l=r ¼ 2:08; clamped–clamped boundary condition ðm; nÞ

Fiber angle 15°

60°

Number of lamina

(1,1) (5,1) (10,1) (15a , 1) a

2

4

6

2

4

6

21.56 21.81 22.57 21.56

26.01 25.93 25.68 25.41

26.73 26.58 26.17 25.49

735.32 785.57 387.93 322.35

752.03 799.05 465.37 399.21

755.13 802.94 478.59 410.82

Lowest circumferential mode corresponding to first axial mode.

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Fig. 10. Characteristic curves of the square of the natural frequency x2 versus the axisymmetric temperature for piezoelectric composite cylindrical shell with l=r ¼ 0:54; clamped–clamped boundary condition and fiber angle (a) 45°, (b) 60° (c) 75° and (d) 90°.

conditions are presented in Fig. 12. Gain provided for the feedback control system was 0.1. It is seen from the Fig. 12(a) and (b) the active damping ratio increases slightly as the temperature increases. When the temperature nears the critical thermal buckling temperature the system is unstable. For a single degree freedom system, f ¼ ðc=cc Þ, where cc ¼ 2mxn . In case, in a system the damping remains same and xn is changed without changing the mass it can be assumed that f will change depending on xn . Even though this argument cannot directly be extended to complicated multi-degree freedom system, to some extent the behavior of f with respect to temperature can be explained. Whenever temperature increases, only the system stiffness matrix changes and hence the frequency of the system decreases with increase in temperature. From Fig. 12 it is seen that at temperatures that are little away from buckling temperature xn decreases and hence f of

the system increases slightly. In contrast, near about the buckling temperature the behavior shifts from this trend and the system becomes unstable. This may be due to influence of buckling phenomenon. One also notices certain deviations, like the curves illustrated for composite shell with fiber angle 45°, Fig. 12(a), the first axial mode of circumferential harmonic 10 does not become unstable. Referring to Fig. 12(b), for composite shell with fiber angle 60°, the first axial mode of circumferential harmonic 1 becomes unstable much before the temperature nears the critical thermal buckling temperature.

7. Conclusions Non-linear effects have been considered due to initial stresses in the semi-analytical finite element formulation

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Fig. 11. Characteristic curves of the square of the natural frequency x2 versus the axisymmetric temperature for piezoelectric composite cylindrical shell with l=r ¼ 2:08; clamped–clamped boundary condition and fiber angle (a) 45°, (b) 60° (c) 75° and (d) 90°.

Fig. 12. Active damping ratios versus axisymmetric temperature. Composite cylindrical shell with l=r ¼ 0:54 and clamped–clamped boundary condition. Fiber orientation are (a) 45° and (b) 60°.

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to study the piezothermoelastic behavior of composite cylindrical shells. Formulation is based on the assumption that the temperature is steady and axisymmetric. The sensors and actuators are convolving cosine shaped. Based on the formulation, detailed studies related to static thermal buckling, vibration characteristics and evaluation of active damping ratios are carried out on piezoelectric composite cylindrical shell. The influence of the fiber angle for the composite lamina and the number of plies for the composite shell laminate were also studied. Numerical studies showed that the best fiber angle for the composite lamina as well as laminates which show high thermal buckling temperature are 60° and 75°. The benefit of increasing the number of plies in the composite laminate is minimal for plies with 15° fiber angle. The critical thermal buckling temperature increases in case of composite laminate with fiber angle 60° when compared to single ply composite shell. But the benefit expected as a result of introducing more number of plies is minimal from the point view of thermal buckling temperature. The nature of variation of square of the natural frequency of the first axial mode associated with few circumferential harmonics with respect to steady state axisymmetric temperature is examined. It is found that the frequency decreases as the temperature increases and becomes zero at critical thermal buckling temperature. The active damping ratio are found to increase as the temperature increases, the order of increase is very small, and the system becomes unstable as temperature nears the critical thermal buckling temperature.

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