Journal of Thermal Stresses, 33: 79–96, 2010Copyright © Taylor & Francis Group, LLCISSN: 0149-5739 print/1521-074X onlineDOI: 10.1080/01495730903409235

FREE VIBRATION OF THERMALLY PRE/POST-BUCKLEDCIRCULAR THIN PLATES EMBEDDED WITH SHAPEMEMORY ALLOY FIBERS

Shi-Rong Li1�2, Wen-Shan Yu2, and R. C. Batra31School of Civil Science and Engineering, Yangzhou University,Yangzhou, Jiangsu, China2Department of Engineering Mechanics, Lanzhou University of Technology,Lanzhou, Gansu, China3Department of Engineering Science and Mechanics, Virginia PolytechnicInstitute and State University, Blacksburg, VA, USA

Axisymmetric deformations of a uniformly heated pre-buckled and post-buckled thincircular plate reinforced with shape memory alloy (SMA) fibers placed in the radialdirection only are studied. Effects of von Kármán’s nonlinearities are incorporated inthe problem formulation. The matrix is assumed to be linear thermoelastic and thethermo-mechanical response of the SMA is modeled by one-dimensional constitutiverelation. By assuming that plate’s deflections can be additively decomposed intothree parts, namely radial displacements of a pre-buckled plate, radial and lateraldisplacements during post-buckling deformations, and infinitesimal radial and lateraldisplacements during vibration of a post-buckled plate, boundary-value problems fordetermining these displacements for plate edges either simply supported or clamped havebeen formulated. The coupled nonlinear differential equations have been numericallysolved by the shooting method that has been verified by good agreement between thepresently computed results with those available in the literature. The dependence ofthe first three frequencies upon the temperature rise, for both pre-buckled and post-buckled plates, has been delineated. Characteristic curves relating the frequency withthe temperature rise for different values of the volume fraction and the pre-strain inthe SMA fibers are exhibited. It is found that reinforcement of an aluminum plate withSMA fibers changes plate’s natural frequencies and enhances its resistance to bucklingdue to temperature rise.

Keywords: Circular plate; Natural frequencies; Pre-buckling and post-buckling; Shape memory alloy;Shooting method

Received 8 January 2009; accepted 16 May 2009.SRLs and WSYs work was supported by the National Natural Science Foundation of China

grant numbers 10872083 and 10602021, and the Fund of the Ministry of Education of China forSupervisors of Doctorate Students (200807310002). RCB’s work was partially funded by the U.S. Officeof Naval Research grants N00014-06-1-0567 and N00014-06-1-0913 to Virginia Polytechnic Instituteand State University. The authors gratefully acknowledge these financial supports. Views expressed inthe paper are those of authors and neither of the funding agencies nor of their institutions.

Address correspondence to Shi-Rong Li, School of Civil Science and Engineering, Yangzhou

79

University, Yangzhou, Jiangsu 225009, China. E-mail: list2000@yahoo.com.cn

80 S.-R. LI ET AL.

INTRODUCTION

Plates are key components in many structural applications. Bucklingand vibration of plates due to thermal loads play important roles in thedesign and analysis of aerospace, off-shore, under-water, nuclear, and electroniccomponents/structures. Static and dynamic responses of plates and shells exposedto thermal environments have been reviewed by Tauchert [1] and Thorton [2].Even though there have been extensive investigations on the thermal buckling ofplates, the dynamic response of post-buckled circular plates has not been studiedthoroughly. The vibration response of flat and curved panels subjected to thermo-mechanical loads has been studied by Librescu et al. [3, 4]. Li and Zhou [5, 6]used the shooting method to analyze vibrations of uniformly heated orthotropiccircular/annular plates. Lee and Lee [7] examined vibrations of pre-buckled andpost-buckled anisotropic plates by using the first-order shear deformation platetheory (FSDT). Free vibration response of a thermally buckled piezo-laminatedcomposite plate has been investigated by Oh et al. [8] by using the finite elementmethod (FEM). Infinitesimal vibrations of a post-buckled functionally graded platehave been studied by Park and Kim [9], and of a statically deformed pre-buckled orpost-buckled plate by Li and Batra [10].

Studies [3–10] on vibration of heated thin plates and shells reveal thatnatural frequencies of a thermally loaded structure are noticeably affected by thetemperature rise. However, effects of temperature rise on free vibration of platesembedded with shape memory alloy (SMA) fibers will be more significant becausethe temperature rise will change the pre-strain in the fiber and may induce aphase transformation in the SMA fibers that will alter their mechanical properties.Many research efforts have contributed to our understanding of the vibration andthe buckling control of plates reinforced with SMA fibers. Rogers et al. [11–13]experimentally and analytically investigated the acoustic and vibration control of anSMA hybrid composite plate made of graphite/epoxy and NiTiNol/epoxy laminas.Ro and Baz [14] studied effects of activating NiTiNol fibers on the bucklingand vibration of SMA reinforced composite plate by the FEM coupled withexperimental observations. Ostachowicz et al. [15, 16] used the FEM to examinein detail effects of embedding SMA actuators on the natural frequencies andthermal buckling of laminated plates. Based on the FSDT, thermal buckling andlinear vibration of pre-buckled and post-buckled SMA fiber-reinforced laminatedcomposite plates were examined by Zhong et al. [17, 18] and Park et al. [19, 20].They used the FEM to analyze the influence of the volume fraction and the initialstrain of SMA fibers on natural frequencies and buckling deflection of plates.It should be noted that they used experimental data to model the recovery stress andYoung’s modulus versus the temperature rise instead of an analytical constitutiverelation for the SMA, and found that the SMA fibers could greatly reduce orcompletely eliminate the post-buckling deflection for appropriate values of thetemperature rise.

Lee et al. [21] used the commercial FE computer code ABAQUS to analyzelaminated composites shells and plates with embedded SMA wires, and showedthat the activation of SMA wires due to heating increases the critical bucklingtemperature and decreases the thermal buckling deformation. Thompson andLoughlan’s extensive experimental and numerical simulations [22, 23] indicate that

PRE/POST-BUCKLED CIRCULAR THIN PLATES 81

recovery forces in pre-strained SMA fibers produced by thermal activation enhancethe buckling temperature for various laminated plates and reduce the out-of-plane displacement of the post-buckled laminates. More recently, Zhang et al. [24]studied vibration of laminated composite plates with embedded SMAs by usingthe Rayleigh-Ritz method. In the aforementioned investigations, the SMA fiber-reinforced composite structures are either beams or rectangular plates.

We study here thermally induced pre-buckling and post-buckling deformationsof a circular plate reinforced with SMA fibers. We use the one-dimensionalconstitutive relation proposed by Brinson [25] to find the recovery stress in the SMAfiber. The plate edge is either simply supported with boundary points restrainedfrom moving in the radial direction or clamped. The strain-displacement relationsincorporate von Kármán nonlinearities and the stress-strain relation is taken to belinear. The nonlinear governing equations are numerically solved with the shootingmethod to analyze the pre-buckling and the post-buckling deformations of the plate.Numerical results presented herein include the thermal post-buckled equilibriumpaths and frequencies of free vibrations of post-buckled configurations.

PROBLEM FORMULATION

We consider a thin composite circular plate of radius b and thickness hcomposed of an isotropic and homogenous matrix, and reinforced shape memoryalloy (SMA) fibers in the radial direction. A cylindrical co-ordinate system �r� �� z�located in the mid-surface of the plate as shown in Figure 1 is used to describeplate’s deformations. The material of plate is considered to be polar orthotropic.We assume that a steady uniform temperature change T is applied to the plate, andthe plate particles at its boundary are restrained from moving within the mid-surfaceso that in-plane thermal stresses are induced due to the temperature change.We study axisymmetric free vibration of the pre-buckled and the post-buckledheated plate by incorporating the von Kármán nonlinearity in the Kirchhoff platetheory. The strain-displacement relations are

�r =�u

�r+ 1

2

(�w

�r

)2

− z�2w

�r2�� =

u

r− z

r

�w

�r� �r� = ��r = 0 (1)

where u�r� t�� w�r� t� are the radial and the transverse displacements of a point onthe mid-surface; t the time; �r and �� the radial and the circumferential strains,respectively; �r� the shear strain.

We assume that the material response for infinitesimal deformations of theelastic composite plate can be adequately described by the following constitutiveequations [17–20]:

�r =Er

1− r��r

[�r + ���r − �r + ��r��T

]+ �sVs (2)

�� =E�

1− r��r

[�� + �rr� − �� + rr���T

](3)

In Eqs. (2) and (3), Er and E� are Young’s moduli in the radial and thecircumferential directions, respectively; r and � the thermal expansion coefficients;

82 S.-R. LI ET AL.

Figure 1 Schematic sketch of the circular plate with SMA fibers embedded in the radial direction.

r� and �r Poisson’s ratios; �T = T − T0 the temperature rise; T0 the initialtemperature; �r and �� normal stresses in the radial and the circumferentialdirections; �s the recovery stress induced by the inverse martensitic transformationof the SMA due to the temperature change �T ; Vs the volume fraction of theSMA fibers taken to be uniform in the plate. The assumptions of Vs being constantand the stress in the SMA affecting only �r greatly simplify the problem. If SMAreinforcements are wires then for Vs to be constant either their diameter will need todecrease with the radial coordinate or the spacing between two adjacent wires willhave to increase. In Eqs. (2) and (3) we have assumed that Ez = Er .

Homogenization techniques for deriving effective properties of compositeswith SMA as reinforcements have been discussed, for example, in [31–35]. Effectivevalues of material parameters in terms of those of the matrix and the SMA given inAppendix A are derived by the mechanics of materials approach.

Using one-dimensional constitutive equations for SMA, the recovery stress ofSMA fibers is given by the following relation [21, 25]:

�s = �E� �− E� 0���0 +�� � s −�� 0� s0 +��T − T0� (4a)

E� � = EA + �EM − EA�� �� � = −�LE� � (4b)

PRE/POST-BUCKLED CIRCULAR THIN PLATES 83

where E� �, �� �, �, � �0 and �L are, respectively, Young’s modulus, the phasetransformation coefficient, the coefficient of thermal expansion, the volume fractionof martensite, the initial axial strain, and the maximum residual strain of SMAfibers. Quantities with the subscript “0” denote their values in the initial stateof SMA. EM and EA are, respectively, Young’s moduli of the SMA in the puremartensite and the pure austenite phases. The parameter defined in the range�0� 1� is divided into the stress-induced part s and the temperature-induced part T .Here we only consider temperature rise.

For T > As and CA�T − Af� < �s < CA�T − As�, � s and T are computedfrom the following relations [25]:

= 02

{cos

[aA

(T − As −

�s

CA

)]+ 1

}(4c)

s = s0 − s0 0

� 0 − � (4d)

T = T0 − T0 0

� 0 − � (4e)

where As and Af denote the austensite transformation start and finish temperaturerespectively, CA equals the slope of the curve of the critical stress for austensite phasetransformation as function of temperature, aA = �/�Af − As�, and T0 is the initialvolume fraction of martensite.

Substitution from Eq. (1) into Eqs. (2) and (3), and integrating their zerothorder and first order moments with respect to the thickness coordinate z over theplate thickness give following expressions for the membrane forces and the bendingmoments:

Nr =∫ h/2

−h/2�rdz = C

[�u

�r+ 1

2

(�w

�r

)2

+ �r

ru− r�1+ ��r��T

]+ �sVsh (5a)

N� =∫ h/2

−h/2��dz = C

[�r

(�u

�r+ 1

2

(�w

�r

)2)+ k

ru− r�k� + �r��T

](5b)

Mr =∫ h/2

−h/2�rzdz = −D

(�2w

�r2+ �r

r

�w

�r

)(6a)

M� =∫ h/2

−h/2��zdz = −D

(�r

�2w

�r2+ k

r

�w

�r

)(6b)

where k = E�/Er = �r/r� is the ratio of elastic moduli, � = �/r is the ratio ofcoefficients of thermal expansion, C = 12D/h2 and D = Erh

3/�12�1− �rr��� arethe in-plane and the flexural rigidities of the plate.

We introduce the following non-dimensional quantities.

�x� U�W� = �r� u� w�/b� � = b/h (7a)

� = 12�2�1+ �r��r�T� � = ��r + �k�/�1+ ��r� (7b)

N ∗ = �sVs�1− �rr��/Er� � = �t/b2��Dr/�h�1/2 (7c)

where �, defined in Appendix A, is the mass density of the homogenized plate.

84 S.-R. LI ET AL.

By neglecting the in-plane and the rotary inertia and using Hamilton’sprinciple, we obtain the following partial differential equations and the associatedboundary conditions governing free vibration of the circular plate in terms of non-dimensional variables [5, 6, 10].

�2U

�x2+ 1

x

�U

�x− k

x2U + �W

�x

�2W

�x2+ 1− �r

2x

(�W

�x

)2

= 1x

[�

12�2�1− ��− N ∗

](8)

�4W

�x4+ 2

x

�3W

�x3− k

x2�2W

�x2+ k

x3�W

�x+ ��− 12�2N ∗�

(�2W

�x2+ 1

x

�W

�x

)+ �2W

��2

= 12�21x

�

�x

[x

(�U

�x+ 1

2

(�W

�x

)2

+ �r

xU

)�W

�x

](9)

U = 0��W

�x= 0�

�3W

�x3+ 1

x

�2W

�x2= 0� at x = c (10a,b,c)

U = 0� W = 0��W

�x+ K

�2W

�x2= 0� at x = 1 (11a,b,c)

The parameter c is a small positive real number introduced to avoid a singularity atx = 0 while computing numerical results [5, 29], and K = 0 and 1/�r , respectively,for the clamped and the simply supported edges. Boundary conditions (10b)and (10c) imply that the transverse shear force Qx = −(

�3W�x3

+ 1x�2W�x2

− 1x2

�W�x

)vanishes

at x = c. In the limit of c approaching zero boundary conditions (10) are satisfiedat the center of a solid circular plate.

GOVERNING EQUATIONS FOR INCREMENTAL DISPLACEMENTS

In order to study infinitesimal vibration and static small deformations of thethermally buckled circular plate we assume that the solution of Eqs. (7)–(10) can beexpressed as [3–6, 10]

U�x� �� = U0�x�+ Us�x�+ Ud�x� �� (12)

W�x� �� = Ws�x�+Wd�x� �� (13)

where U0�x� is the in-plane radial displacement of the pre-buckled plate,Us�x� and Ws�x� are the radial and the circumferential displacements of thethermally post-buckled plate, Ud�x� �� and Wd�x� �� are infinitesimal displacementssuperimposed on the thermally buckled configuration of the plate.

The displacement U0 is the solution of the boundary-value problem

d2U0

dx2+ 1

x

dU0

dx− k

x2U0 =

1x

[�

12�2�1− ��− N ∗

](14)

U0�c� = U0�1� = 0 (15)

PRE/POST-BUCKLED CIRCULAR THIN PLATES 85

whose analytical solution is [5, 6, 10]

U0�x� =

B

1− k

[�c − c−

√k�x

√k − �c − c

√k�x−

√k

c√k − c−

√k

+ x

]k �= 1

0 k = 1

(16)

where B = ���1− ��/12�2 − N ∗�. In the limit c → 0, Eq. (16) gives [10]

U0�x� =

B

1− k�x − x

√k� k �= 1

0 k = 1(17)

For � = 1 and N ∗ = 0 Eq. (14) with boundary conditions given by Eq. (15) has atrivial solution U0 = 0. For � �= 1 and N ∗ = 0, the problem reduces to that studiedin [10]. Thus the present work differs from our earlier work [10] in N ∗ not beingzero because of the stress induced in the SMA.

The incremental displacements, Us�x� and Ws�x�, from the buckled to thepost-buckled configurations are determined by solving the following non-linearboundary-value problem [10]:

d2Us

dx2+ 1

x

dUs

dx− k

x2Us +

dWs

dxd2Ws

dx2+ 1− �r

2x

(dWs

dx

)2

= 0 (18)

d4Ws

dx4+ 2

x

d3Ws

dx3− k

x2d2Ws

dx2+ k

x3dWs

dx+ ��− 12�2N ∗�

(d2Ws

dx2+ 1

x

dWs

dx

)

= 12�21x

ddx

[x

(dU0

dx+ dUs

dx+ 1

2

(dWs

dx

)2

+ �r

x�U0 + Us�

)dWs

dx

](19)

Us = 0�dWs

dx= 0�

d3Ws

dx3+ 1

x

d2Ws

dx2= 0� at x = c (20)

Us = 0� Ws = 0�dWs

dx+ K

d2Ws

dx2= 0� at x = 1 (21)

Deformations of the SMA affect Us�x� and Ws�x� through the presence of U0 inEq. (19).

For studying infinitesimal vibration of the post-buckled plate, we neglectterms nonlinear in Ud�x� �� and Wd�x� ��. Substitution from Eqs. (12)–(13) intoEqs. (8)–(11), using Eqs. (18)–(21), we obtain the following linear homogeneouspartial differential equations for finding Ud�x� �� and Wd�x� ��.

�2Ud

�x2+ 1

x

�Ud

�x− k

x2Ud +

dWs

dx�2Wd

�x2+ �Wd

�x

d2Ws

dx2+ 1− �r

x

dWs

dx�Wd

�x= 0 (22)

�4Wd

�x4+ 2

x

�3Wd

�x3− k

x2�2Wd

�x2+ k

x3�Wd

�x+ ��− 12�2N ∗�

(�2Wd

�x2+ 1

x

�Wd

�x

)+ �2Wd

��2

86 S.-R. LI ET AL.

= 12�21x

�

�xx

[(dU0

dx+ dUs

dx+ 1

2

(dWs

dx

)2

+ �r

x�U0 + Us�

)�Wd

�x

+(�Ud

�x+ dWs

dx�Wd

�x+ �r

xUd

)dWs

dx

](23)

Ud = 0��Wd

�x= 0�

�3Wd

�x3+ 1

x

�2Wd

�x2= 0� at x = c (24)

Ud = 0� Wd = 0��Wd

�x+ K

�2Wd

�x2= 0� at x = 1 (25)

Thus deformations of SMAs influence Ud�x� �� and Wd�x� �� because U0 and N ∗

appear in Eq. (23).We assume that Eqs. (22)–(25) have harmonic solutions [3–6, 10]

Ud�x� �� = ��x� cos����� Wd�x� �� = ��x� cos���� (26)

where

� = �

√�hb4

Dr

is the non-dimensional frequency and � the dimensional frequency. Substitutionfrom Eq. (26) into Eqs. (22)–(25) yields the following ordinary differential equationsfor amplitudes ��x� and ��x�:

d2�

dx2+ 1

x

d�dx

− k

x2�+ dWs

dxd2�

dx2+ d�

dxd2Ws

dx2+ 1− �r

x

dWs

dxd�dx

= 0 (27)

d4�

dx4+ 2

x

d3�

dx3− k

x2d2�

dx2+ k

x3d�dx

+ ��− 12�2N ∗�(d2�

dx2+ 1

x

d�dx

)− �2�

= 12�21x

ddx

x

[(dU0

dx+ dUs

dx+ 1

2

(dWs

dx

)2

+ �r

x�U0 + Us�

)d�dx

+(d�dx

+ dWs

dxd�dx

+ �r

x�

)dWs

dx

](28)

� = 0�d�dx

= 0�d3�

dx3+ 1

x

d2�

dx2= 0� at x = c (29)

� = 0� � = 0�d�dx

+ Kd2�

dx2= 0� at x = 1 (30)

We note that Eqs. (27)–(30) give amplitudes of free vibration of SMA fiber-reinforced composite plate with initial radial and lateral displacements U0�x�+ Us�x�and Ws�x�, respectively. However, before solving Eqs. (27)–(30), we need first tosolve the non-linear and coupled Eqs. (18)–(21) for Us�x� and Ws�x�. If let Us�x� =Ws�x� = 0, or the plate has not buckled, then Eqs. (27)–(30) can be reduced togoverning equations for the linear vibration of the pre-buckled plate.

PRE/POST-BUCKLED CIRCULAR THIN PLATES 87

Table 1 Material properties of the SMA fibers

CM = 8MPa/�C Mf = 9�C � = 0�55MPa/�C �L = 0�067CA = 13�8MPa/�C Ms = 18�4�C s = 10�26× 10−6/�C s = 0�33�crs = 100MPa As = 34�5�C �s = 6500 kg/m3 EM = 26�3× 103 MPa.

�crf = 170MPa Af = 49�C EA = 67× 103 MPa

NUMERICAL RESULTS AND DISCUSSION

Because of our inability to analytically solve the nonlinear boundary-valueproblem defined by Eqs. (18)–(21) we solve it numerically with the shooting methodthat replaces the two-point boundary-value problem by a sequence of initial-valueproblems. As described in [5, 6, 10, 26], values of functions at the initial point,x= c, are estimated to start the computations. These are iterated upon with modifiedvalues obtained by the secant method until the prescribed boundary conditions atthe final point x = 1 are satisfied. It should be noted that the constitutive Eq. (4)is nonlinear and an explicit form for the recovery stress cannot be obtained; it isdetermined by the Newton–Raphson method.

While solving the problem numerically, we set initial temperature T0 = 20�C,� = b/h = 30, and take c = 0�001. The matrix of the composite plate is aluminumfor which �m = 2800kg/m3, Em = 70GPa, m = 0�3, m = 23�4× 10−6/�C, andmaterial properties the NiTiNol SMA, taken from [25], are listed in Table 1.

The numerical algorithm has been verified by comparing computed results fora few problems with those available in the literature; e.g., see Tables 1–3 of [10].Results presented emphasize effects of the initial pre-strain and the volume fractionVs of SMA fibers on the pre- and the post-buckling deformations of the plate.

Thermal Buckling

The post-buckling equilibrium paths, in terms of the maximum deflectionA=W�0� and the thermal load �, of clamped and simply supported plates for knownvalues of the SMA volume fraction, Vs, are plotted in Figure 2 with �0 = 0�04 as

Figure 2 Equilibrium paths of post-buckled circular plate for specified values of Vs ��0 = 0�04�.

88 S.-R. LI ET AL.

Figure 3 Equilibrium paths of post-buckled circular plate for different values of �0 �Vs = 10%�.

the initial strain in the SMA fiber. It is clear that an increase in the SMA volumefraction noticeably reduces the post-buckling deformation of the composite plate,and enhances its critical buckling temperature. For a high SMA volume fraction, thethermally buckled plate can revert to its unbuckled configuration in the temperaturerange of the inverse martensite phase transformation, and the plate will again bucklewith further increase in the temperature.

For Vs = 10%� 20% and various value of �0, results exhibited in Figures 3and 4 delineate effects of the thermal load � on the central deflection A. Theseevince that, for a simply supported plate, the initial strain in the SMA fibers hasa little influence on the critical buckling temperature but a significant effect on thethermally induced post-buckling deformation because the inverse martensite phasetransformation of SMA fibers occurs during the post-buckling region. However,for the clamped plate with Vs = 20%, the critical buckling temperature can beenhanced by increasing the pre-strain in the SMA fibers. This is because an increasein �0 increases the temperature at which the inverse martensite phase transformationis completed. It implies that the buckling occurs after the martensite phase

Figure 4 Equilibrium paths of post-buckled circular plate for different values of �0 �Vs = 20%�.

PRE/POST-BUCKLED CIRCULAR THIN PLATES 89

Figure 5 Dependence of the natural frequencies of the pre/post-buckled plate upon the temperaturerise.

has completely been transformed into the austenite phase. Therefore, in general,an increase in either �0 or Vs will decrease the post-buckling deflection of the platedue to the higher recovery stress in the SMA fibers. These results are qualitativelysimilar to those for rectangular plates obtained by Zhong [17, 18], Lee [21] and Park[19, 20].

Vibration

By simultaneously solving boundary-value problems defined by Eqs. (18)–(21)and (27)–(30), frequencies and the corresponding mode shapes of a thermallypre-buckled and post-buckled SMA fiber reinforced circular plate with theboundary either simply supported or clamped are obtained. Figures 4–7 exhibit thedependence of the first three natural frequencies upon the thermal load parameter

Figure 6 For �0 = 0�04, dependence of the natural frequencies of the pre/post-buckled clampedcircular plate upon the temperature rise.

90 S.-R. LI ET AL.

Figure 7 For �0 = 0�04, dependence of the natural frequencies of the pre/post-buckled simplysupported circular plate upon the temperature rise.

� with and without the SMA fibers in the pre-buckling and the post-bucklingregions. From the results shown in Figure 5, we conclude that the first three lowestfrequencies of a pre-buckled plate without SMA fibers decrease monotonicallywith an increase in the temperature, and, as expected, the fundamental frequencyapproaches zero at the critical temperature. In the post-buckling region, thefrequency of the plate increases with an increase in the temperature because thelarge deflection of the post-buckled plate increases the bending stiffness of the plate[10, 19, 20].

However, for the clamped plate reinforced with the SMA fibers, frequenciesdecrease first, then increase or adjust, and then decrease in the pre-bucklingregion for Vs = 25% as presented in Figure 6. This phenomenon results fromthe increase of the bending stiffness due to the recovery stress of SMA with

Figure 8 For �0 = 0�04 and different values of Vs , fundamental frequencies versus the temperature risefor the circular plate.

PRE/POST-BUCKLED CIRCULAR THIN PLATES 91

Figure 9 For �0 = 0�04 and different values of Vs , an expanded view of the fundamental frequenciesversus the temperature rise for the circular plate presented in Figure 8.

Figure 10 For Vs = 20% and different values of �0, the dependence of the first three lowest frequenciesupon the non-dimensional temperature rise for the clamped circular plate.

92 S.-R. LI ET AL.

initial strain [17–20]. In addition, as seen from Figure 6, the temperature range ofpre-buckling deformations can be enlarged because of the temperature activationof the embedded SMA fibers. For relatively small volume fractions of SMA fibers,frequencies change in both the pre-buckling and the post-buckling regions, but forrelatively large volume fractions of SMA fibers, frequencies adjust only in the pre-buckling region. Frequencies of the simply supported SMA-reinforced plate onlydecrease in the pre-buckling region as presented in Figure 7. This is because thestart temperature of the inverse martensite transformation is larger than the criticalbuckling temperature for the simply supported plate; thus frequencies adjustmentincluding the control of buckling as discussed in the previous section can only beseen in the post-buckling regime.

Figure 8 displays variation of the fundamental frequency of the clamped andthe simply supported plates in the pre- and the post-buckled regions for specifiedvalues of Vs. It is evident that with an increase in Vs the fundamental frequency isreduced in the post-buckling region because the increased recovery force depressesthe post-buckling deflection (Cf. Fig. 2). In Figure 9 we have plotted an enlarged

Figure 11 For Vs = 20% and different values of �0, the dependence of the first three lowest frequenciesupon the non-dimensional temperature rise for the simply supported circular plate.

PRE/POST-BUCKLED CIRCULAR THIN PLATES 93

view of the local part of the frequency-temperature curves in the pre-buckling regionshown in Figure 8. It is clear that the frequency decreases with an increase in thevalue of Vs, which primarily is due to the increase of the plate weight and thenonexistence of the recovery stress in SMA fibers as noted by Park et al. [19].For Vs = 20%, we have exhibited in Figures 10 and 11 the influence of the pre-strainin the SMA fibers on the first three lowest frequencies. From Figure 10 we seethat the adjustment for the frequency of clamped plate can be realized in the pre-buckling region because the start temperature of inverse martensite transformationis less than the critical buckling temperature. Along with the increase in values of thepre-strain of SMA the pre-buckling region is extended. Nevertheless, as mentioned,SMA fibers embedded in the simply supported plate are activated in the post-buckling region. The variation of frequencies of the plate with an increase in theSMA volume fraction, as shown in Figure 11, is quite interesting.

CONCLUSIONS

Free vibrations of circular plates reinforced with SMA (NiTiNol) fibers inthe radial direction and the plate heated uniformly have been numerically analyzedwith the shooting method when the plate has or has not buckled and the plateboundary is either simply supported or clamped. Effects of the volume fraction andthe pre-strain in the SMA fibers on the thermal post-buckling deflection and thenatural frequencies have been delineated. For a simply supported plate points onthe boundary are restrained from moving radially. It is found that the embeddingof SMA fibers in a circular plate increases the critical buckling temperature anddecreases the post-buckling deflection. However, it has a minimal effect on thecritical buckling load but decreases the post-buckling deflection of the simplysupported circular plate. This is because the start temperature of inverse martensitephase transformation of the NiTiNol is less than the critical buckling temperatureof the plate. The recovery force produced by the pre-strained SMA fibers whenstimulated by heating significantly increases the critical buckling temperature for aclamped circular plate, and the recovery force is enhanced by increasing values ofboth the initial strain and the volume fraction of SMA fibers.

For several values of the pre-strain and the volume fraction of SMAfibers, the three lowest natural frequencies of SMA fiber-reinforced circular plateboth in the pre-buckling and the post-buckling regions have been computed.The frequencies increase in the pre-buckling region for the clamped plates anddecrease it in the post-buckling region for the simply supported plate. This is dueto the increase in the weight of the plate and the decrease in the deflection causedby the SMA fibers [19, 20]. Thus the thermal buckling and the vibration responsecan be efficiently controlled by adjusting values of the volume fraction of SMA, thepre-strain in the SMA fibers and the temperature rise.

APPENDIX A

Adopting the mechanics of materials approach, material parameters for thecomposite are assumed to be related to those of its constituents by the followingrelations [17–20]:

Er = EmVm + EsVs = Em�1− Vs�+ EsVs (A-1)

94 S.-R. LI ET AL.

E� =EmEs

EmVs + EsVm

= EmEs

EmVs + Es�1− Vs�(A-2)

r� = mVm + sVs = m�1− Vs�+ sVs (A-3)

r� = �rk0� k = E�/Er (A-4)

r =EmmVm + EssVs

Er

= Emm�1− Vs�+ EssVs

Er

(A-5)

� = mVm + sVs = m�1− Vs�+ sVs (A-6)

� = �mVm + �sVs = �m�1− Vs�+ �sVs (A-7)

where subscripts ‘m’ and ‘s’ indicate quantities for the matrix and the SMA fiber,respectively.

REFERENCES

1. T. R. Tauchert, Thermally Induced Flexure, Buckling and Vibration of Plates,Applied Mechanics Review, vol. 44, pp. 347–360, 1991.

2. E. A. Thorton, Thermal Buckling of Plates and Shells, Applied Mechanics Review,vol. 46, pp. 485–506, 1993.

3. L. Librescu, W. Lin, M. P. Nemeth, and J. H. Starnes, Jr., Vibration of GeometricallyImperfect Panels Subjected to Thermal and Mechanical Loads, J. Spacecraft Rockets,vol. 33, pp. 285–291, 1996.

4. L. Librescu and W. Lin, Vibration of Thermomechanically Loaded Flat and CurvedPanel Taking into Account Geometric Imperfections and Tangential Edge Restraints,Int. J. Solids and Structures, vol. 34, pp. 2161–2181, 1997.

5. S.-R. Li and Y.-H. Zhou, Shooting Method for Non-Linear Vibration and ThermalBuckling of Heated Orthotropic Circular Plates, J. Sound and Vibration, vol. 248,pp. 379–386, 2001.

6. S.-R. Li, Y.-H. Zhou, and X. Song, Non-linear Vibration and Thermal Buckling of anOrthotropic Annular Plate with a Centric Rigid Mass, J. Sound and Vibration, vol. 251,pp. 141–152, 2002.

7. D. M. Lee and I. Lee, Vibration Behaviors of Thermally Post-Buckled AnisotropicPlates Using First-Order Shear Deformation Plate Theory, Computers and Structures,vol. 63, pp. 371–378, 1997.

8. I. K. Oh, J. H. Han, and I. Lee, Post-Buckling and Vibration Characteristics ofPiezolaminated Composite Plate Subjected to Thermo-Piezoelectric Loads, J. Sound andVibration, vol. 233, pp. 19–40, 2000.

9. J. S. Park and J. H. Kim, Thermal Postbuckling and Vibration Analyses ofFunctionally Graded Plates, J. Sound and Vibration, vol. 289, pp. 77–93, 2006.

10. S. R. Li, R. C. Batra, and L. S. Ma, Free Vibration of Orthotropic Circular Plates withThermal Post-Buckled Deformation, J. Thermal Stresses, vol. 30, pp. 43–57, 2007.

11. C. A. Rogers, Active Vibration and Structural Acoustic Control of Shape MemoryAlloy Hybrid Composites: Experimental Results, J. Acoustic Society of America, vol. 88,pp. 2803–2811, 1990.

12. C. A. Rogers, C. Liang, and C. R. Fuller, Modeling of Shape Memory Alloy HybridComposites for Structural Acoustic Control, J. Acoustic Society of America, vol. 89,pp. 210–220, 1991.

13. C. A. Rogers, C. Liang, and J. Jia, Structural Modification of Simply SupportedLaminated Plates Using Shape Memory Alloy Fibers, Computers and Structures, vol. 38,pp. 569–580, 1991.

PRE/POST-BUCKLED CIRCULAR THIN PLATES 95

14. J. Ro and A. Baz. Nitinol-Reinforced Plates: Part I–III, Composites Engineering, vol. 5,pp. 61–106, 1999.

15. W. Ostachowicz, M. Krawczuk, and A. Zak, Natural Frequencies of a MultilayerComposite Plate with Shape Memory Alloy Wires, Finite Element in Analysis andDesign, vol. 32, pp. 71–83, 1999.

16. W. Ostachowicz, M. Krawczuk, and A. Zak, Dynamic and Buckling of aMultilayer Composite Plate with Embedded SMA Wires. Composite Structures, vol. 48,pp. 163–167, 2000.

17. Z. W. Zhong, Reduction of Thermal Deflection and Random Response of CompositeStructures with Embedded Shape Memory Alloy at Elevated Temperature, PhD Thesis,Old Dominion University, Virginia, 1998.

18. Z. W. Zhong, Buckling and Post-Buckling of Shape Memory Alloy Fiber ReinforcedComposite Plates, Buckling and Post-Buckling of Composite Structures, 41/PVP, vol. 293.ASME, pp. 115–132, 1994.

19. J.-S. Park, J.-H. Kim, and S.-H. Moon, Vibration of Thermally Post-BucklingComposite Plates Embedded with Shape Memory Alloy Fibers, Composite Structures,vol. 63, pp. 179–188, 2004.

20. J.-S. Park, J.-H. Kim, and S.-H. Moon, Thermal Post-Buckling and FlutterCharacteristics of Composite Plates Embedded with Shape Memory Alloy Fibers,Composites Part B: Engineering, vol. 36, pp. 627–636, 2005.

21. H.-J. Lee, J.-J. Lee, and J.-S. Huh, A Simulation Study in the Thermal BucklingBehavior of Laminated Composite Shells with Embedded Shape Memory Alloy Wires,Composite Structure, vol. 47, pp. 463–469, 1999.

22. S. P. Thompson and J. Loughlan, Adaptive Post-Buckling Response of CarbonFiber Composites Plates Employing SMA Actuators, Composite Structures, vol. 38,pp. 667–678, 1997.

23. S. P. Thompson and J. Loughlan, The Control of the Post-Buckling Response in ThinComposite Plates Using Smart Technology, Thin-Walled Structures, vol. 36, 231–263,2000.

24. R. X. Zhang et al., Vibration Characteristics of Laminated Composite Plates withEmbedded Shape Memory Alloys, Composite Structures, vol. 74, pp. 389–398, 2006.

25. L. C. Brinson, One-Dimensional Constitutive Behavior and Shape Memory Alloys:Thermo-Mechanical Derivation with Non-Constant Material Functions and RedefinedMartensite Internal Variable, J. Intelligent Material System and Structures, vol. 4,pp. 229–243, 1993.

26. L.-S. Ma and T.-J. Wang, Nonlinear Bending and Post-Buckling of a FunctionallyGraded Circular Plate under Mechanical and Thermal Loadings, Int. J. Solids andStructures, vol. 40, pp. 3311–3330, 2003.

27. K. K. Raju and G. V. Rao, Thermal Post-Buckling of Circular Plates, Computers andStructures, vol. 18, pp. 1179–1182, 1984.

28. U. S. Gupta and A. H. Ansari, Effect of Elastic Foundation on Asymmetric Vibrationof Polar Orthotropic Linearly Tapered Circular Plates, J. Sound and Vibration, vol. 254,pp. 411–420, 2000.

29. C. L. Huang and H. S. Wolker, Non-Linear Vibration of Hinged Circular Plate witha Concentric Rigid Mass, J. Sound and Vibration, vol. 126, pp. 9–17, 1988.

30. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd Ed.,McGraw-Hill, New York, 1959.

31. B. Jiang and R. C. Batra, Micromechanical Modeling of a Composite ContainingPiezoelectric and Shape Memory Alloy Inclusions, J. Intelligent Materials and Systems,vol. 12, pp. 165–182, 2001.

96 S.-R. LI ET AL.

32. M. Cherkaoui, Q. P. Sun, and G. Q. Song, Micromechanics Modeling of Compositeswith Ductile Matrix and Shape Memory Alloy Reinforcement, Int. J. Solids andStructures, vol. 37, pp. 1577–1594, 2000.

33. D. C. Lagoudas, J. G. Boyd, and Z. Bo, Micromechanics of Active Composites withSMA Fibers, ASME J. Engineering Materials and Technology, vol. 116, pp. 337–347,1994.

34. Z. K. Lu and G. J. Weng, A Two-Level Micromechanical Theory for a Shape MemoryAlloy Reinforced Composite, Int. J. Plasticity, vol. 16, pp. 1289–1307, 2000.

35. K. Tanaka, S. Kobayashi, and Y. Sato, Thermomechanics of TransformationPseudoelasticity and Shape Memory Effect in Alloys, Int. J. Plasticity, vol. 2, pp. 59–72,1986.