Damping in microscale modified couple stress thermoelastic circular Kirchhoff plate resonators

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1 Available at Appl. Appl. Math. ISSN: Vol., Issue (Deceber 7), pp Applications and Applied Matheatics: An International Journal (AAM) Daping in icroscale odified couple stress theroelastic circular Kirchhoff plate resonators * Corresponding author Rajneesh Kuar, Shaloo Devi * and Veena Shara Departent of Matheatics Kurukshetra University Kurukshetra, India. rajneesh_kuk@rediffail.co Departent of Matheatics & Statistics Hiachal Pradesh University Shila, India shalooshara673@gail.co; veena_ath_hpu@yahoo.co Received: June 4, 7; Accepted: Septeber 7, 7 Abstract The vibrations of circular plate in odified couple stress theroelastic ediu using Kirchhoff- Love plate theory has been presented. The basic equations of otion and heat conduction equation for Lord Shulan (L-S, 967) theory are written with the help of Kirchhoff-Love plate theory. The theroelastic daping of icro bea resonators is studied by applying noral ode analysis ethod. The solutions for the free vibrations of plates under claped, siply supported and free boundary conditions are obtained. The analytical expressions for theroelastic daping of vibration and frequency shift are obtained for generalized couple stress theroelastic and coupled theroelastic plates. Nuerical results with the help of MATLAB prograing software in case of silicon aterial has been presented. A coputer algorith has been constructed to obtain the nuerical results. The theroelastic daping and frequency shift with varying values of length and thickness for claped, siply supported and free boundary conditions in the absence and presence of couple stress are presented graphically. Coparisons are ade with different odes of vibrations and also with and without couple stress paraeter. Soe special cases are obtained in the present proble. The theroelastic daping have any applications in (sensors) resonators in detecting Infrared (IR) and iaging in addition to cheical and biological agent sensing and their teperature dependence in design/construction of precision theroeters and counications. 94

2 AAM: Intern. J., Vol, Issue (Deceber 7) 95 Keywords: Modified couple stress theory; Kirchhoff-Love plate theory; Noral ode analysis; Theroelastic daping; Frequency shift MSC No.: 65N5; 74B5; 74B. Introduction The classical couple stress elasticity has been observed to describe the Cosserat sizedependent effects [Mindlin (96, 963), Toupin (96)], which is based on Cosserat continuu theory. However, as it involves four aterial constants for isotropic elastic aterials where two of the are separate aterial length scale paraeters, it is a very difficult task to experientally deterine the icro-structural aterial length scale paraeter. Yang et al. () developed couple stress based strain gradient theory for elasticity. The concept of representative volue eleent was introduced and the force and couple applied to a single aterial particle were defined. They developed a new set of equilibriu relations for a syste of aterial particles to account for the rotations of these aterial particles and then results were generalized to the couple stress theory of continuu. By the introduction of a higher order equilibriu condition, the arbitrary nature of couples in the classical couple stress theory was resolved without the use of rigid vector attachent condition, as was used in the icropolar theory Eringen (966). When an elastic solid is put in otion, it is taken out of equilibriu, having an excess of kinetic and potential energy. The conjugation of strain field to a teperature field provides an energy dissipation echanis that allows the syste to relax back to equilibriu. This process of energy dissipation is called theroelastic daping. The existence of theroelastic daping process by considering the transverse vibration of hoogeneous and isotropic thin bea was constructed by Zener (937, 938). Theroelastic daping originates fro theral currents generated due to the contraction/expansion of elastic edia. The bending of the reed causes dilations of opposite signs to exist on the upper and lower halves. While, one side is copressed and heated, the other side is stretched and cooled. Consequently, a transverse teperature gradient is produced due to the presence of finite theral expansion. The teperature gradient generates local heat currents, which cause increase of the entropy of the reed and lead to energy dissipation. Circular plates are coon eleents in any sensors and resonators Bao and Jiang (998). A icro-resonator based high sensor array to be used as Infrared sensors was presented by Vig et al. (996). These sensors can be used for infrared detection, iaging, cheical and biological agent sensing, while they are not suitable for precision frequency control applications due to their extreely high sensitivity of ass loading. Theroelastic daping in single-crystal silicon and silicon nitride icro-resonators at roo teperature was observed by Yasuura et al. (999). Srikar and Senturia () investigated theroelastic daping in fine-grained polysilicon flexural bea resonators. Nayfeh and Younis (4) developed analytical expressions for the quality factors of icroplate of general shapes due to theroelastic daping. Theroelastic daping of the n-plane vibration of thin silicon rings of rectangular cross-section was presented by Wong et al. (6). A new odel of Kirchhoff plate in odified couple stress theory was constructed by Tsiatas (9). In this odel, the static analysis of isotropic icro-plates with arbitrary shape containing only one internal aterial length scale paraeter which can capture the size effect. Sun and Tohyoh (9) studied the theroelastic daping effect on the vibration of circular plate

3 96 Rajneesh Kuar et al. and gives the derivation of theroelastic equations for a circular plate under out-of-plane flexural vibration. The basic equations of coupled theroelastic theory are constructed by Sun and Saka () for out of plane vibration of a circular plate resonators. Shara and Shara () studied the daping in icro-scale generalized theroelastic circular plate resonators under claped plate and siply-supported plate. Fang et al. (3) investigated the proble of theroelastic daping in the axisyetric vibration of circular icroplate resonators with two diensional heat conduction. A new odel of Kirchhoff plate in odified couple stress theory was developed by Shaat et al. (4), which studies the effects of surface energy and icrostructure on the plate rigidity and deflection. Forced vibration analysis of a icroplate on the basis of odified couple stress theory and Kirchhoff plate theory was presented by Sisek et al. (5). Darijani and Shahdadi (5) developed a new non-classical shear deforation plate odel in odified couple stress theory including two unknown functions. Gao and Zhang (6) constructed a non-classical Kirchhoff plate odel by applying odified couple stress theory, surface elasticity theory and two paraeter elastic foundation. Reddy et al. (6) discussed the proble of functionally graded circular plates with odified couple stress theory by using finite eleent ethod. Chen and Wang (6) investigated a odel for coposite lainated Reddy plate of new odified couplestress theory and global local theory. This paper deals with the study of vibrations of circular plate resonators in odified couple stress theroelastic ediu using Kirchhoff-Love plate theory. The theory of generalized (non-classical) theroelasticity given by Lord and Shulan (967) has been applied to solve the proble. The effects of theroelastic daping and frequency shift on vibrations of circular plate resonators are presented analytically and shown graphically with varying values of length and thickness for claped, siply-supported and free plates in the absence and presence of couple stress. Soe special cases are also given.. Basic equations Following Yang et al. () and Rao (7), the constitutive equation, the equations of otion and the equation of heat conduction for Lord-Shulan (L-S, 967) theory in a odified couple stress theroelastic odel in the absence of body forces and body couples are Constitutive relations tij ekk ij e ij - ekijlk,l - Tij, () ij ij, ij i, j j,i, i eipquq, p, () Equation of otion... T, 4 u 4 u u (3)

4 AAM: Intern. J., Vol, Issue (Deceber 7) 97 Equation of heat conduction where t ij Kronecker s delta, KT cet T., t t u (4) are the coponents of stress tensor, e ij and are the coponents of strain tensor, coponents of couple-stress, t e ijk are La e, constants, is alternate tensor, ij ij is are the 3 t. Here are the coefficients of linear theral expansion respectively, T is the teperature change, is the couple stress paraeter, is syetric curvature, is the rotational vector. u is the displaceent vector, is the density, is the Laplacian operator, is del operator. K is the coefficient of the theral conductivity, is the specific heat at constant strain, is the reference teperature c e assued to be such that i T / T T. Here is the theral relaxation tie. ij 3. Forulation of the proble We consider a thin couple stress theroelastic circular plate with unifor thickness h and radius b. The origin of the Cartesian coordinate syste r,, z is taken at the centre of the plate. To siply the proble, we apply the fundaental Kirchhoff-Love plate hypothesis for the coupling plate vibrations following Huang (5) for the present case: (i) (ii) (iii) Noral stress can be neglected relative to the principal stresses, i.e., tzz. The rectilinear eleent noral to the iddle surface before deforation reains perpendicular to the strained surface after deforation and their elongation can be neglected, i.e. erz e z. For sall deflection vibration, the deforation along the iddle surface can be neglected, i.e., ezz. t zz In equilibriu conditions, the plate is unstrained, unstressed and continues at unifor environental teperature T everywhere. We define the displaceent coponents ur,, z, t, vr,, z, t, wr,, z, t and teperature T r,, z, t. The displaceent coponents are given by w r,, t z w r,, t ur,, z, t z, vr,, z, t, wr,, z, t wr,, t, r r (5) The strain coponents are taken as u w, r z r r (6)

5 98 Rajneesh Kuar et al. u v w w z, r r rr r (7) v u w r = r + v = -z. r r r r (8) and stress coponents are established by the constitutive relation as: t t E ( ) T, (9) r r T E ( ), r TT () tr, r () The bending and torsion oents are defined as: h h r h r h r M t zdz + dz, () h h h h r M t zdz dz, M h t zdz h r r + r, h h dz (3) (4) w w w TM T w w w M r D h, r r r r d r r r r (5) w w w TM T w w w M D h, r r r r d r r r r (6) M r 3 h w w w h, 6 r r r rr (7) The equations for shear force resultants are rm rm r M r r M Qr M, Q M r. r r r r (8)

6 AAM: Intern. J., Vol, Issue (Deceber 7) 99 The equation of otion (force equilibriu z in the direction) is given by Qr Q Qr w + h. r r r t (9) The equation of otion for icro plate with syetry about where D Eh 3 4 T D h w M T h y axes is taken as E w, d t () is the flexural rigidity of the plate, E 3 is Young s odulus, = is the Poisson ratio and q x, t represents the load acting along the thickness direction. The equation of heat conduction for L-S theory is given by T w z t t x () K T K τ c et Tz, For convenience, we define the non-diensional paraeters as: x, z, u, w t T M E x, z, u, w, τ, t, T, M,. ' ' ' ' ' ' ' ' T T L L L T dt h () Using non-diensional quantities defined by equation () on equations () and (), we get h 4 ETT h L h L w w MT, D D D t (3) T cel L T τ T z w, z t t K K (4) 4. Proble solution Assuing the tie haronic vibrations as: i,, t n i t n,,,,,. (5) w r t W r e T r z t r z e n n i Here, t is the frequency of the plate and W re the ode shape of the displaceent. Here,,,3,... denote the nuber of nodal diaeters and n,,3,... the nuber of

7 93 Rajneesh Kuar et al. nodal circles. In general, the frequency is coplex. The real part gives us new eigen-frequency of the plate in the presence of theroelastic coupling, theral relaxation I g provides tie and echanical relaxation ties. The agnitude of iaginary part us the attenuation of vibrations. Substituting equation (5) in equations (3) and (4), we obtain, Re h *4 ETT h L * h L W MT W, D D D (6) and * cel L * i i z W,. (7) z K K 5. Theral field on the thickness direction We assue that the theral gradient of the bea is very sall as copared to that along its thickness direction, ie.., x z as in Shara and Kaur (4). Therefore, the equation (7) take the for, z L z K t * p, W (8) where h cl e t t T K h p, i iτ, M d r, z z dz. (9) Let us assue that there is no heat flow across the upper and lower surfaces of the bea. Then, z at z h. (3) With the use of this condition, the general solution of equation (8), can be written as

8 AAM: Intern. J., Vol, Issue (Deceber 7) 93 L sin pz r z z W Kp p pcos t *,, (3) Substituting equation (3) in equation (9), the theral oent is given by 3 dl h t * MT f p W, (3) Kp where 4 ph ph f p tan. ph 3 (33) Using equation (3) in equation (6), yields: D W a W, (34) *4 where W p W, (35) *4 4 E T h L d a,,,. 5 T t h h L 4 D f p a p Kp D D D D (36) For axisyetric vibration of circular plate, the solution of equation (35) is W r A J pr A Y pr A I pr A K pr 3 4,. (37) The coefficients through A and A 4 allowed values of are deterined by the boundary conditions. In case of liitation of W r at the plate center, we get A A4, then 3, p W r A J pr A I pr (38) Here, J and kind. I represents the Bessel and odified Bessel functions of order n and of first 6. Boundary Conditions Let us consider following set of boundary conditions: (i) Claped plate

9 93 Rajneesh Kuar et al. W dw,. dr (39) r a ra (ii) Siply supported plate Equation (4), with the aid of (34) reduces to W W M (4), T T. ra ra W ra (4), W. ra (iii) Free plate W W W, r r r r ra (4) W W r r r r r W ra. (43) 7. Characteristic Equations Substituting equation (38) in the boundary conditions (39), (4) and (4) and (43), we get (I) Claped plate A J pa A I pa (44) 3, A p J pa J pa A p I pa I pa pa pa 3, (45) For nontrivial solutions of equations (44) and (45), we obtain J pa I pa J pa J pa I pa I pa pa pa. (46) (II) Siply supported plate J pa I pa. J pa I pa (47) (III) Free plate

10 AAM: Intern. J., Vol, Issue (Deceber 7) 933 ' ' ' ' ' ' pa J pa pa J pa J pa pa I pa pa J pa J pa pa I pa pa I pa I pa pa I pa pa I pa I pa The solution of characteristic equations (46), (47) and (48) as, where the values of q,,,3; n,,3 are described in Table for claped, siply supported and free plates. 8. Frequency shift and Daping pa q Now the vibration frequency of the plate in the presence of theroelastic coupling and theral relaxation tie is given by, (48) q a (49) 4, D D q h f p. a D (5) where q. (5) a and following Shara (), we can replace first order, we obtain f p with f and expand equation (5) upto h ie TT h L t f p. 4D 4Kp D (5) The theral gradients in the plane of cross-section along the thickness direction of the plate are uch larger than those along its length and hence so that x This iplies that i cel p t, (53) K i * cel s * p p e, p, s, tan. K (54) Replacing with in equations (55) and (56), we obtain

11 934 Rajneesh Kuar et al. and * p p cos i sin, (55) * * cel s * p, s, tan. K (56) The frequency is coplex in nature and hence we take R I R I i, Re, Ig, (57) and where and R h I f R,. 4 f I D f 6 cos sin tan sinh 6cos R, 3 * * p cos cos h h ph f I 6 cos tan sin sinh 6sin, 3 * * p cos cosh h ph (58) (59) (6) * where ph cos, tan and is taken fro equation (5). The frequency shift and daping in a theroelastic plate are taken fro Shara () R s. (6) and Q I. (6) R 9. Particular cases (i). Coupled theroelastic (CT) plate In the absence of theral relaxation tie, we obtain

12 AAM: Intern. J., Vol, Issue (Deceber 7) 935 * cel * * p p i, p, s, =,, ph. K Accordingly, equations (6) and (6) becae f R 3 * ph cos cos h 6 sin sinh, (63) and f I cos cosh * * ph ph sin sinh. (64) (ii). If couple stress paraeter, equation (6) reduces to R i E T h L I i E Th L T,. 4 t T 4 t f R f I Kp D Kp D (65) where 6 cos sin tan sinh 6 cos tan sin sinh 6cos 6sin 3 f R, f I. 3 * * * * p cos cos h cos cosh h ph ph ph (66). Nuerical results and discussion For the purpose of nuerical coputations, we have prepared atheatical odel with silicon aterial. The physical data of the proble are taken fro Sun and Saka (). The values of theroelastic daping and frequency shift of vibration odes (,) and (,) with thickness h and length L in a claped plate, siply-supported plate and free plate have been coputed in the absence and presence of couple stress. The nuerical coputations have been carried out with the help of MATLAB software. The coputer siulated results have been presented graphically in figs. -. Table. Following [Shara and Shara (), Rao (7)], the values of q,,,3; n,,3 for claped, siply-supported and free plates n Claped plate Siply supported plate Free plate q q q q n q n q 3n n q n q 3n n q n 3n ,

13 936 Rajneesh Kuar et al. Physical data for Silicon aterial: Quantity Silicon aterial Unit 65.9 Kg - s -. Kg -3 E T c e K t d H L K 73 J Kg - K - 56 W - K K - N Sec -.. Sec Daping (=)(,) (=5)(,) (=)(,) (=5)(,). Daping Thickness Figure. Theroelastic daping Q a claped plate. of vibration odes (,) and (,) with thickness h in

14 AAM: Intern. J., Vol, Issue (Deceber 7) (=)(,) (=5)(,) (=)(,) (=5)(,) 3 Daping Thickness Figure. Theroelastic daping a siply supported plate. Q of vibration odes (,) and (,) with thickness h in (=)(,) (=5)(,) (=)(,) (=5)(,).5 Daping Thickness Figure 3. Theroelastic daping Q a free plate. of vibration odes (,) and (,) with thickness h in

15 938 Rajneesh Kuar et al. 3.5 (=)(,) (=5)(,) (=)(,) (=5)(,) Daping Length Figure 4. Theroelastic daping claped plate. Q of vibration ode (,) and (,) with length L in a 4 (=)(,) (=5)(,) (=)(,) (=5)(,) Daping Length Figure 5. Theroelastic daping Q siply supported plate. of vibration ode (,) and (,) with length L in a

16 AAM: Intern. J., Vol, Issue (Deceber 7) (=)(,) (=5)(,) (=)(,) (=5)(,).5 Daping Length Figure 6. Theroelastic daping free plate. Frequency shift Q of vibration ode (,) and (,) with length L in a (=)(,) (=5)(,) (=)(,) (=5)(,) Frequency shift Thickness Figure 7. Frequency shift s of vibration odes (,) and (,) with thickness h in a claped plate.

17 94 Rajneesh Kuar et al (=)(,) (=5)(,) (=)(,) (=5)(,) 6 Frequency shift Thickness s Figure 8. Frequency shift of vibration odes (,) and (,) with thickness h in a siply supported plate (=)(,) (=5)(,) (=)(,) (=5)(,) Frequency shift Thickness Figure 9. Frequency shift s of vibration odes (,) and (,) with thickness h in a free plate.

18 AAM: Intern. J., Vol, Issue (Deceber 7) (=)(,) (=5)(,) (=)(,) (=5)(,) Frequency shift Length Figure. Frequency shift plate. s of vibration ode (,) and (,) with length L in a claped (=)(,) (=5)(,) (=)(,) (=5)(,) Frequency shift Length Figure. Frequency shift s of vibration odes (,) and (,) with length L supported plate. in a siply

19 94 Rajneesh Kuar et al..5 (=)(,) (=5)(,) (=)(,) (=5)(,) Frequency shift Length Figure. Frequency shift plate. s of vibration odes (,) and (,) with length L in a free Daping Figure shows the theroelastic daping with thickness h in a claped plate of vibration odes (,) and (,). It is observed that the daping factor decreases onotonically in the considered region of thickness for all cases. The values of daping factor is greater in the absence of couple stress and saller for presence of couple stress. Figure depicts the theroelastic daping with thickness h in a siply supported plate of vibration odes (,) and (,). It is noticed that the behaviour and variation of daping factor are sae but difference between their values. The values of daping factor of vibration ode (,) is observed to have ore value than that of vibration ode (,) for absence and presence of couple stress. Figure 3 represents the theroelastic daping with thickness h in a free plate of vibration odes (,) and (,). The value of daping factor is larger for sall values of thickness and reains stationary for higher value of thickness in the assued range for all cases of couple stress. Figure 4 depicts the theroelastic daping with length L in a claped plate of vibration odes (,) and (,). The value of theroelastic daping increases with increasing value of length. It is observed that the value of daping factor is greater in case of vibration ode (,) in coparison to vibration ode (,) for, 5. Figure 5 represents the theroelastic daping with length L in a siply supported plate of vibration odes (,) and (,). The values of daping factor increases soothly with increase in length for all cases of couple stress and also different vibration odes in the considered range. The daping factor of vibration ode (,) have greater value than that of vibration ode (,) in the presence of couple stress, whereas its value of vibration ode (,) is ore in coparison with vibration ode (,) for absence of couple stress. Figure 6 shows the theroelastic daping with length L in a free plate of vibration odes (,) and (,). It is observed that

20 AAM: Intern. J., Vol, Issue (Deceber 7) 943 the value of daping factor increases with increasing values of length L. Moreover, daping factor have ore value in the vibration ode (,) in coparison with vibration ode (,) for absence and presence of couple stress. Frequency shift Figure 7 represents the frequency shift of vibration ode (,) and (,) with thickness h in a claped plate. It is clear that the frequency shift of vibration odes (,) and (,) first decreases for sall values of thickness and then increases for higher value of thickness for the considered range. The value of frequency shift of vibration ode (,) has greater value of vibration ode (,) for. Figure 8 represents the frequency shift of vibration ode (,) and (,) with thickness in a siply supported plate. The behaviour and variation are alost sae but difference between their values. It is evident fro the figure that the value of frequency shift of vibration ode (,) has ore value in absence of couple stress than that of presence of couple stress. Also, frequency shift of vibration ode (,) is observed greater value for in coparison with. Figure 9 represents the frequency shift of vibration ode (,) and (,) with thickness h in a free plate. The value of frequency shift initially decreases and then increases soothly in the assued range of thickness. It is noticed that the values of frequency shift of vibration odes (,) and (,) have greater value for absence of couple stress in coparison with presence of couple stress.,5 h 5 Figure represents the frequency shift of vibration ode (,) and (,) with length L in a claped plate. It is observed fro the figure that the value of frequency shift increases with increasing value of length. The value of frequency shift is observed to have greater value in case of vibration ode (,) than that of vibration ode (,) for all cases of couple stress. Figure represents the frequency shift of vibration ode (,) and (,) with length L for a siply supported plate. The values of frequency shift increases onotonically with increase in the value of length for all cases of couple stress and vibration odes (,) and (,). It is clear fro the figure that value of frequency shift has ore for absence of couple stress in coparison with presence of couple stress for vibration ode (,). Siilarly, frequency shift of vibration ode (,) has less value for presence of couple stress than that of absence of couple stress. Figure represents the frequency shift of vibration ode (,) and (,) with length L for a free plate. It is noticed that the behaviour and variation are sae for vibration odes (,) and (,). The values of frequency shift for vibration ode (,) have greater value in case of vibration ode (,) in the absence of couple stress, whereas opposite behaviour is observed for presence of couple stress.. Conclusions This paper devoted to the study of vibrations of circular plate in odified couple stress theroelastic ediu in the context of Kirchhoff-Love plate theory and Lord-Shulan theroelasticity theory. The atheatical expressions for theroelastic daping of vibration and frequency shift are obtained in the absence and presence of couple stress for (,) and (,) vibration odes of the plate. Daping factor and frequency shift with varying values of length and thickness are shown graphically to show the effect of couple stress for vibration odes (,) and (,) with claped plate, siply supported plate and free plate. It is concluded fro the figures that the daping factor decreases with increase in the value of thickness, whereas its value increases with increasing value of thickness for both cases of couple stress and both vibration odes (,) and (,). The value of frequency shift initially

21 944 Rajneesh Kuar et al. decreases and then increase in the value of thickness but the value of frequency shift increases with increase in the value of length in the considered range. It is also concluded that the theroelastic daping factor and frequency shift of vibration odes (,) and (,) with increase in the values of thickness attains larger value in the absence of couple stress than that of presence of couple stress for claped, siply-supported and free plates. The theroelastic daping factor and frequency shift for absence and presence of couple stress for vibration ode (,) have ore proinent value in coparison with vibration ode (,) for claped, siply-supported and free plates. The results of this proble ay be useful for Infra-Red (IR) detections and iaging in addition to cheical and biological agent sensing. REFERENCES Bao, G. and Jiang, W. (998). A heat transfer analysis for quartz icroresonator IR sensors, Int. J. Solids and Structures, Vol. 35, pp Chen, W. and Wang, Y. (6). A odel of coposite lainated Reddy plate of the globallocal theory based on new odified couple-stress theory, Mechanics of Advanced Materials and Structures, Vol. 3, No. 6, pp Darijani, H. and Shahdadi. A.H. (5). A new shear deforation odel with odified couple stress theory for icroplates, Acta Mech., Vol. 6, No. 8, pp Eringen, A.C. (966). Linear theory of icropolar elasticity, J. Math. Mech., Vol. 5, pp Fakhrabadi, S.M.M. (7). Application of Modified Couple Stress Theory and Hootopy Perturbation Method in Investigation of Electroechanical Behaviors of Carbon Nanotubes, Advances in Applied Matheatics and Mechanics, Vol. 9, No., pp Fang, Y., Li, P. and Wang, Z. (3). Theroelasic daping in the axisyetric vibration of circular icroplate resonators with two diensional heat conduction, J. Theral Stresses, Vol. 36, pp Gao, X.L. and Zhang, G.Y. (6). A non-classical Kirchhoff plate odel incorporating icrostructure, surface energy and foundation effects, Continuu Mech. Therodyn., Vol. 8, pp Huang, C.H. (5). Transverse vibration analysis and easureent for the piezoceraic annular plate with different boundary conditions, Journal of Sound and Vibration, Vol. 83, pp Lord, H.W. and Shulan, Y. (967). A generalized dynaical theory of theroelasticity, J. of Mech. and Phys. of Solids, Vol. 5, pp Mindlin, R.D. and Tiersten, H.F. (96). Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., Vol., pp Mindlin, R.D. (963). Influence of couple-stresses on stress-concentrations, Exp. Mech., Vol. 3, pp. 7. Nayfeh, A.H. and Younis, M.I. (4). Modeling and siulations of theroelastic daping in icroplates, J. Microechanics and Microengineering, Vol. 4, pp Rao, S.S. (7). Vibration of continuous systes. John Wiley & Sons, Inc. Hoboken, New Jersey. Reddy, J.N., Roanoff, J. and Loya, J.A. (6). Nonlinear finite eleent analysis of functionally graded circular plates with odified couple stress theory, European Journal of Mechanics-A/Solids, Vol. 56, pp. 9 4.

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