Large Amplitude Flexural Vibration of the Orthotropic Composite Plate Embedded with Shape Memory Alloy Fibers

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1 Chinese Journal of Aeronautics 0(007) 5- Chinese Journal of Aeronautics Large Amplitude Flexural Viration of the Orthotropic Composite Plate Emedded with Shape Memory Alloy Fiers Ren Yongsheng a, *, Sun Shuangshuang a College of Mechanical and Electronic Engineering, Shandong University of Science and echnology, Qingdao 6650, China School of Electro-mechanical Engineering, Qingdao University of Science and echnology, Qingdao 6606, China Received January 007; accepted July 007 Astract he free and forced viration of large deformation composite plate emedded with shape memory alloy (SMA) fiers is investigated. A thermo-mechanical constitutive equation of SMA proposed y Brinson et al. is employed and the constitutive equations for evaluation of the properties of a hyrid SMA composite laminate are otained. Based on the nonlinear theory of symmetrically laminated anisotropic plates, the governing equations of flexural viration in terms of displacement and stress functions are derived. he Galerkin method has een used to convert the original partial differential equation into a nonlinear ordinary differential equation, which is then solved with harmonic alance method. he numerical results show that the relationship etween nonlinear natural frequency ratio and temperature for the nonlinear plate has similar characteristics compared with that of the linear one, and the effects of temperature on forced response ehavior during phase transformation from Martensite to Austenite are significant. he effects of the volume fraction of the SMA fier, aspect ratio and free viration amplitude on the dynamical ehavior of the plate are also discussed. Keywords: smart materials; large-deflection plate; composite; nonlinear viration Introduction * In recent ten years, great progress has een made on the research of shape memory alloy (SMA) reinforced smart structure systems. hrough the design of SMA intelligent composite structure systems with highly integrated sensors and actuators, the control of complex flexile structures with rigorous weight limitations such as aeronautic structures has ecome possile. For example, the uckling and viration characteristics of SMA reinforced composite plate and eam structures have een greatly improved when compared to traditional composite structures, this makes SMA e widely used in uck- *Corresponding author. el.: address: renys@sdust.edu.cn ling and viration control [-]. For the structures with high strength and large flexiility used in aircraft design, the geometric nonlinear effect resulted from large deformation cannot e ignored. For instance, the shapes of the panels of the supersonic aircraft will e changed due to the large thermal deflections induced y aerodynamic heating, which will affect the aerodynamic characteristics and reduce the flight performance of the aircraft. In order to control such kind of nonlinear structures, it is necessary to know the effect of geometric nonlinearity on the viration of structures. However, most of the researches on SMA reinforced composite plates and eams are ased on the classic small deflection theory and only a few are concerning with the large deflection nonlinearity issues.

2 6 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5- Based on von Karman plate theory, Chu L C [] estalished the nonlinear finite element model for the SMA fier reinforced composite large-deflection plate and studied the suppression of the flutter of panels of aircraft. Zou Jing [] derived the incremental finite element motion equation for the nonlinear composite laminate emedded with SMA fiers on the asis of the virtual work principle, and the ending, thermo-uckling and post-uckling issues of the SMA reinforced composite laminate under transverse loading were discussed. Dano M L et al. [5] studied the effect of SMA on the snapthrough characteristics of the nonlinear unsymmetric fier-reinforced composite laminate with largedeflections. hey estalished the approximate theory to analyze the snap-through characteristics of the unsymmetric fier-reinforced composite laminate activated y SMA wires, where the mechanical properties of the laminate are predicted with the assumed strain-displacement field, Rayleigh-Ritz method and the virtual work principle, and the law of snap-through varying with the temperature was otained through solving relevant simultaneous equations and the equation descriing SMA properties. Park J S et al. [6-7] estalished the finite element model of composite laminates and supersonic plates under nonlinear viration and nonlinear flutter y using the von Karman plate theory, the one-order shearing deformation plate theory and the first-order piston theory. he oundary prolems of thermopost-uckling nonlinear viration and flutter were then investigated for the SMA fier reinforced composite plate under thermal and aerodynamic loading. Cho M et al. [8] studied the deformation of the nonlinear composite plate with two-way shape memory effect with the theory of the one-order shearing deformation plate with large deflections and the thermo-mechanical constitutive equation proposed y Lagoudas et al. Based on the aove literature review, we realize that: most of the current researches on the geometric nonlinearity of the SMA fier (or layer) reinforced composite plate are limited to using the finite element numerical method [-7], and little work has een reported to use the nonlinear elastic theory to get the analytical viration solution for the SMA reinforced nonlinear anisotropic laminates; aout the description of the mechanical ehavior of SMA, some are ased on the expression of the approximate experimental fitting [], the others use the approximate data derived from the curves of SMA [6-7], and the well-developed and practical constitutive equation of SMA ased on the phenomenal theory such as Brinson s equation [,9] is hardly used; the effect of SMA fiers on the nonlinear viration of composite plates still needs to e investigated further. In this paper, the free and forced viration of the SMA reinforced composite laminates with large deflections will e studied. he thermo-mechanical constitutive equation of SMA proposed y Brinson et al. [] and the mixture theory for evaluating the properties of laminates are employed to estalish the constitutive equation of the SMA reinforced composite laminates with large deflections. Based on the nonlinear theory of symmetrically laminated anisotropic plates, the transverse viration equation and the compatile equation will e derived in terms of the transverse deflection and the stress function. he Galerkin approximate method and the harmonic alance method (HBM) will e used to solve the Duffing s differential equation and to study the effect of the content of SMA fiers, the nonlinear viration amplitude and the excitation force amplitude on the natural frequency and steady-state response of the system. Basic heories. he constitutive equation of the anisotropic laminate with emedded SMA fiers As the SMA reinforced anisotropic lamina has the following off-axis stress-strain relation [0] x x x rx ( k) ( k) yqij y Q ij y ry () xy xy xy r xy

3 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5-7 the constitutive equation of the SMA reinforced laminate can e written as 0 N A B r M B D M Mr N N () 0 where and denote the strain vector and the curvature vector of the mid-plane, respectively. he in-plane forces and moments induced y SMA (denoted y the suscript r) and temperature (denoted y the suscript ) are respectively expressed as N M N M N N N h ( ) ( ) ( ) k k k r r r x y xy dz h M M M h ( k) ( k) ( k) rx ry rxy zdz h N N N r rx ry rxy r rx ry rxy x y xy h h h h ( k) ( k) ( k) x y xy M M M x y xy ( k) ( k) ( k) x y xy dz zdz he A, B and D in Eq.() are expressed as A A A6 B B B6 A A A A, B B B B 6 6 A6 A6 A 66 B6 B6 B 66 () D D D6 D D D D 6 () D6 D6 D 66 where each element in Eq.() is defined as h ij ij ij ij A, B, D Q, z, z dz (5) h where Q ij represents the off-axis elastic constants of the SMA fier hyrid lamina. he recovery stresses of SMA and the stresses induced y temperature of the k lamina after coordinate transformation have the following form, respectively. ( k ) m n mn rx rvs ry n m mn 0 0 rxy mn mn m n ( k ) Q x Q Q6 m n y Q Q Q 6 n m mn( ) xy Q Q Q (6) where Q ij (i =,, 6; j =,, 6) denotes the elastic constant of the composite medium. m and n are the cosine and sine functions of the ply angle in each lamina, respectively. From Eq.(), we have 0 M B D M Mr (7) 0 A N B A ( N N ) (8) where A A, B A B. Sustituting Eq.(8) into Eq.(7), we get M B ( r ) A N B A N N D M Mr BA N ( DBB ) BA ( N N ) M M With the following expression r r r (9) BA BA ( A B) ( B ) (0) Eq.(9) can e rewritten as M ( B ) N ( B ) ( N Nr ) D M M () where D D BA B. By expanding Eq.(8), we have the following expression 0 x A A A6 0 0 y A A A6 0 A xy 6 A6 A 66 N B x B B 6 x Ny B B B6 y Nxy B6 B6 B66 xy A A A6 Nx Nrx A A A6 Ny Nry A6 A6 A66 Nxy Nrxy r ()

4 8 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5- where w,, w,, w, ; and w is x xx y yy xy xy the transverse deflection of the laminate.. ransverse viration equations For convenience, the stress function ( x, y) is introduced as a separate variale and the in-plane forces are expressed in terms of ( x, y) as N, N, N () x, yy y, xx xy, xy For the symmetric orthotropic laminate, we Qij h Q ij have Bij 0, Aij, Dij Dij, M M r = h 0 and Q Q with the expression of Q as E E 0 Q E E 0 () 0 0 G where Q is the matrix consisting of elastic constants of the orthotropic lamina with, E E ; E i, ij and G ij (i =, ; j =, ) denote the engineering elastic constants of the SMA fier hyrid composites which are calculated according to the mixture method proposed y Zhong Z W et al. [] (see appendix A). According to the characteristics of the symmetric orthotropic laminates, we have the following expressions C A C, A h Eh h h Eh h C C66 A, A66 h Eh h Gh A A66 ( ) Eh G h h G E h Also, Eq.(8)and Eq.() are simplified as (5) 0 A N A ( N Nr ) (6) M D (7) hen the transverse viration equation in terms of the transverse deflection w and the stress function is written as Dw Dw D w, xxxx, xxyy, yyyy w, tt q L ( w, ) 0 (8) Eh E h where D, D, D D D, Gh D, and q represent the density and the transverse excitation force of the plate respectively. he arithmetic operator L * (w, ) in Eq.(8) is expressed as L ( w, ) w w w (9), xx, yy, yy, xx, xy, xy he compatile equation is written as x, yy y, xx xy, xy w, xy w, xxw, yy (0) Sustituting Eq.(6) into Eq.(0) and considering that N and N r are independent of the coordinates, we get the following supplemental equation h, xxxx, xxyy, yyyy L ( w, w ) 0 () he von Karman s nonlinear strain-displacement relations are 0 0 x u, x w, x 0 0 y v, y w, y xy u, y v, x w, xw, y () where u 0, v 0 are in-plane displacements of the plate in x and y directions, respectively. Replacing the items in the first two equations in Eq.(6) with Eq.() and integrating them for x and y, we can otain the displacements in the mid- plane. he in-plane displacement oundary conditions can e expressed as 0 0 a a, yyd, x xa xa a a a w, xdx aa( N x Nrx) a u u A x A A( Ny Nry ) y y, xxd, y v v A y A, d ( r ) w y y A N x N x A ( Ny Nry ) 0 () ()

5 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5-9 It should e noted that Eq.(8) and Eq.() are the asic equations of the SMA reinforced orthotropic laminates with large deflections in nonlinear viration, and Eq.() and Eq.() are the equations with in-plane unmovale constraints.. Galerkin approximate solution For the simply-supported plate, one-item approximate solution is taken as x y wxyt (,, ) hf( t)cos cos (5) a Sustituting the aove solution into Eq.(), we have, xxxx, xxyy, yyyy x y h f () t (cos cos ) a a Assume the general solution of Eq.(6) is p c (6) (7) where, p and c denote the inhomogeneous particular solution and the homogeneous general solution of Eq.(6), respectively. Assume x y p Ccos Csin (8) a Sustituting it into Eq.(6) produces h f () t a h f () t C, C a A A aking c as (9) c A x Ay (0) Sustituting x y p c Ccos Csin Ax Ay a into the displacement oundary conditions Eq.() and Eq.(), we have A h A f () t N y a A 8 A A h A f () t N x A a 8 A () where A A, N x = N x N rx, N y = N y A N ry. Using the Galerkin method after sustituting Eq.(5) and Eq.(7) into Eq.(8), we get a Dw, xxxx Dw, xxyy D w a, yyyy x y w, tt ql ( w, ) cos cos dxdy 0 a () By integrating the aove equation and making some simplifications, the nonlinear differential equation with respect to time t is otained. d f( t) h D D dt a a hf () t D Nx Ny a () t A a A a A A A a A ( a) h f a q sin( t ) A A a () If some non-dimensional quantities are introduced as elow Eh a t, Nrx N rx, Nry N ry Eh Eh N N, N N x x y y Eh Eh q q Eh Eq.() will ecome a non-dimensional equation. d f ( ) E d E G Nrx Nx 6 E ()

6 0 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5- E Nry Ny f( ) 6 E E E E E E E f ( ) qcos t (5) E where. E. he recovery stress of the constrained SMA fiers Based on the one-dimensional model of SMA proposed y Brinson [9] and assuming all SMA fiers are fully constrained, we can get the following expressions for the recovery stress of SMA E ( ) E ( ) ( ) r 0 ( 0) 0 As s s 0 0 s ( 0) s0 ( As ) As Af ( Af ) Af (6) where denotes the Martensite fraction; Es ( ) denotes the elastic modulus of SMA, represents the thermal elastic modulus, represents temperature and 0 is the reference temperature, the suscript 0 denotes initial conditions, A s and A f denote the start and finish temperatures of Austenite under stress, s is the Martensite fraction induced y stress. he first expression in Eq.(6) is used for SMA in the initial Martensite state, while the third expression is used for SMA in 00% Austenite state, and the second one is used for SMA in the phase transformation state from Martensite to Austenite. he phase transformation coefficient ( ) and the elastic modulus E s ( ) can e expressed respectively as ( ) L E s ( ) Es( ) EA ( EM EA) (7) he Austenite start temperature A s and finish under stress have the following temperature expressions A f aaas A( 0 0) As aa A Af aaas A{[ EA E( 0)] 0 ( ) } ( a ) 0 s0 0 0 A A he two constants in Eq.(6) are (8) 0 ( As 0) EA Es( 0) 0 ( 0) s0 ( Af As ) (9) he dynamic equations of SMA transformed from Martensite to Austenite are 0 r cos aa( As ) C A s0 s s0 ( 0 ) ( 0 ) 0 (0) 0 s0 0 a A aa, A Af As CA a M am, M Ms Mf CM where M s, M f, A s and A f denote the Martensite start, Martensite finish, Austenite start and Austenite finish temperatures respectively, denotes the Martensite fraction induced y temperature, C M and C A are phase transformation constants. Numerical Results and Discussions Using HBM, the natural frequency equation and the steady-state frequency-response equation in forced viration of the system are otained as follows: N KN A () L KL ( KL ) A KNA q () where E G KL E 6 E Nrx Nx Nry Ny

7 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5- K E E N 6 E E E E E E () where A, A are the free and forced viration amplitude respectively, L and N are linear and nonlinear natural frequencies respectively, is excited frequency. In the present work the symmetric Nii/graphite/epoxy laminate [0/90/0/90/0] s with the thickness h = 0.00 m is used for simulation, where the thickness of each layer is h k = 0.00/0 m (k =,,, 0). he material parameters used in this simulation are given elow. () Graphite/epoxy [] Em 55(.50 ) GPa Em 8.07(.70 ) GPa Gm.55( ) GPa m 586 kg / m 6 m (.50 ) (/ C) 6 m 0.0 (0.0 ) (/ C) ref, ref 0 C, m 0. () SMA fiers EA 67 GPa, EM 6. GPa, Ms 8. C Mf 9 C, As.5 C, Af 9 C CM 8 MPa / C, CA.8 MPa / C, 0.55 L 0.067, 0 0, 0 0 C, s , , s 0., s 6 50 kg / m 6 s 0.60 (/ C) he simulation results are shown in the following figures. Fig. gives the free-viration frequency ratio of the Nii/graphite/epoxy symmetric laminate varying with temperature, where 5, /h = 0, V s = 0. and A =.0. It can e seen that the natural frequency of the nonlinear laminate is larger than that of the linear one due to the increased in-plane stiffness and the frequency ratio of the nonlinear laminate to the linear one shows ascending, descending and then ascending trend with the increasing of temperature. his situation results from the variation of the recovery stress of the fully constrained Nii fiers in the laminate. When temperature is etween 0 and 0 C, as Nii fiers are not actuated, the recovery stress r is zero. But the in-plane compressive forces induced y temperature exist in the laminate, which results in the descending of L as temperature is increasing; When temperature lies within the range of 0 C and A s, L, still decreases with the increasing of temperature. hough Nii fiers are actuated in this temperature range, yet new Austenite has not produced and r is very small. he laminate is still in in-plane compression; When temperature is etween A s and A f, r increases oviously with the increasing of temperature due to the phase transformation from Martensite to Austenite and the total in-plane forces of the laminate ecome tensile, so L increases oviously with the increasing of temperature; When temperature is higher than A f, the curve shows the same trend as that in the temperature range of 0 C and A s. Because Martensite in Nii fiers are fully transformed into Austenite and the in-plane forces induced y temperature ecome dominant forces again that lead to the total compressive in-plane forces in the laminate. Fig. he free-viration frequency ratio of Nii/graphite/ epoxy symmetric laminate varying with temperature. (= 0.5, /h = 0, V s = 0., A =.0) It should e pointed out that L is existing in the denominator of Eq.(), so it shows an opposite trend with that of r as shown in Fig.. Fig.(a) shows the free-viration frequency ratio varying with temperature for Nii/ raphite/epoxy laminates with two non-dimensional viration amplitudes. We can see that the smaller the non-dimensional viration amplitude is, the flatter the

8 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5- curve ecomes. he effects of volume fractions of Nii on the free-viration frequency ratio are shown in Fig.(). From this figure we can see that the volume fraction of Nii can improve the free-viration frequency ratio. And the higher the volume fraction of Nii is, the more wide the curve varying range is. his figure also demonstrates that the natural frequency of the linear laminate ecomes more and more ovious with the increase of volume fractions of Nii, which means that the less effects of the geometric nonlinearity of the laminate will e when the volume fraction of Nii fiers is higher. responses. It can e seen that the temperature nearly has no effects on the shapes of the frequency-response curves, ut it has certain effects on the positions of the framework curves, that is the framework curves shift rightwards with the increasing of temperature. Among the three typical cases, the framework curves will shift rightwards with the largest range when Nii fiers are in the phase transformation state from Martensite to Austenite. his is due to the large increase of r and results in the large increase of the in-plane stiffness, which corresponds to the conclusion derived from Fig.. he frequency-response curves of the Nii hyrid laminate are shown in Fig.6 with the aspect ratio of =0.5, =0.50 and =.00. It can e seen that the frequency-response curves are sensitive to the aspect ratio. he framework curve of the frequency-response curves will shift leftwards with a large range when is increased while the viration amplitudes are also increased oviously. (a) he effects of non-dimensional viration range A Fig. Frequency responses of the laminate emedded with 00%-Austenite Nii fiers under the excitations with different amplitudes. (=00, =0.5, V s =0.) () he effects of volume fractions of ini fiers V s Fig. he free-viration frequency ratio of Nii/graphite/ epoxy symmetric laminate varies with temperature. (=0.5, /h=0) Under forced viration, the steady-state frequency responses of the laminate emedded with Nii fiers in 00% Austenite, initial 00% Martensite and phase transformation from Martensite to Austenite states are shown in Fig., Fig. and Fig.5 respectively. In each figure, three amplitudes of the excitation forces with q 5, q 0 and q 50 are considered to oserve their effects on the frequency Fig. Frequency responses of the laminate emedded with initial 00%-Martensite Nii fiers under the excitations with different amplitudes. (=0, =0.5, V s =0.)

9 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5- Fig.5 Frequency responses of the laminate emedded with Nii fiers in the phase transformation state from Martensite to Austenite under the excitations with different amplitudes. (=70, =0.5, V s =0.) frequency ratio of the laminate under nonlinear free viration increases with the increasing of volume fractions of Nii fiers and the non-dimensional viration amplitudes. () Under the forced viration, the framework curve of the frequency-response curves of the laminate will shift rightwards with the largest range when Nii fiers are in the phase transformation state from Martensite to Austenite due to the corresponding largest in-plane stiffness. () he aspect ratio of the laminate has a dramatic effect of the frequency-response curves. he framework curve of the frequency-response curves will shift leftwards with a large range and the viration range will also e increased oviously. References [] Birman V. Review of mechanics of shape memory alloy structures. Appl Mech Rev 997; 50: [] Ren Yongsheng. Progress of SMA intelligent composites. Journal of Composite Materials 999; 6(): -7. [in Chinese] Fig.6 Frequency responses of the laminate emedded with 00%-Austenite Nii fiers at three different aspect ratios. (=00, V s =0., q 0 ) Conclusions Based on the nonlinear theory of the symmetric orthotropic elastic laminates and the Brinson s SMA model, the natural frequency equation under free viration and the steady-state frequency-response equation under forced viration are estalished for the SMA reinforced composite laminates with large deflections y using the Galerkin approximate method and HBM. he following conclusions can e drawn: () he trend of the frequency ratio of Nii/ graphite/epoxy symmetric laminate under nonlinear free viration varies with temperature is similar to that of the recovery stresses of fully constrained Nii fiers when actuated. And this ehavior ecomes more ovious when the volume fraction of Nii fiers is higher. () At the same actuation temperature, the [] Chu L C. Integrated intelligent structures for suppressing static aerothermoelastic deformations and flutter of panels. AFOSR-R , 995. [] Zou Jing. he constitutive relation of shape memory alloy and the analysis y the finite element method of shape memory alloy reinforced composite laminated plates. PhD thesis, Huazhong University of Science and echnology; 999. [in Chinese] [5] Dano M L, Hyer M W. SMA-induced snap-through of unsymmetric fier-reinforced composite laminates. Int J Solids and Structures 00; 0: [6] Park J S, Kim J H, Moon S H. Viration of thermally post-uckled composite plates emedded with shape memory alloy fiers. Composite Structures 00; 6: [7] Park J S, Kim J H, Moon S H. hermal post-uckling and flutter characteristics of composite plates emedded with shape memory alloy fiers. Composites: Part B 005; 6: [8] Cho M, Kim S. Structural morphing using two-way shape memory effect of SMA. Journal of Solids and Structures 005; : [9] Brinson L C. One-dimensional thermomechanical constitutive relations for shape memory alloy: thermomechanical derivation with non-costant material functions and redefined martensite internal variale. J Intelli Mater Syst and Struct 99; : 9-.

10 Ren Yongsheng et al. / Chinese Journal of Aeronautics 0(007) 5- [0] Zhong Z W, Chen R R, Mei C. Buckling and postuckling of shape memory alloy fier-reinforced composite plates. In: Noor A K, editor. Buckling and postuckling of composite structures. New York: ASME, 99; 5-. [] Zhong Z W, Mei C. Finite element viration analysis of composite plates with emedded shape memory alloy fiers at elevated temperatures. Design Engineering echnical Conference. New York: ASME, 98(): [] Duan B, aw K M, Goek S, et al. Analysis and control of large thermal defection of composite plates using shape memory alloy. SPIE_s 7th International Symposium on Smart Structures and Materials. Newport Beach, CA, 000(99): Biographies: Ren Yongsheng Born in 956, he received B.S. and M.S. from aiyuan University of echnology in 98 and from Southeast University in 989 respectively. He received his Ph.D. from Nanjing University of Aeronautics and Astronautics in 99. He is currently a professor in College of Mechanical and Electronic Engineering, Shandong University of Science and echnology. His research interests include nonlinear dynamics, smart materials, viration and shock control and aeroelasticity, etc. He has pulished several scientific papers in different periodicals. renys@sdust.edu.cn Sun Shuangshuang Born in 97, she received B.S. and M.S. from Qingdao University of Science and echnology in 996 and 999 respectively. She received her Ph.D. from Shanghai Jiaotong University in 00. She is currently an associate professor in College of Electro-mechanical Engineering, Qingdao University of Science and echnology. Her research interests include nonlinear dynamics and smart materials. sunkira@sohu.com Appendix A: he Elastic Constants of Nii Fier Hyrid Composites he elastic constants are calculated according to the following mixture theory [] EmEs E Em( Vs) EV s s, E E V E ( V ) m s s s G G G V V G m s, m ( s ) s s GmVs Gs ( Vs ) E E ( V ) E V s m m s s s s s, ( s) E ( V ) V m s s s N m ( V ) V ( z z ) 0 m s s s k k i where the suscript m denotes the composite medium, and the suscript s denotes SMA. E, G and denote the tension-compression elastic modulus, the shearing elastic modulus and the possion ratio respectively. and denote thermal expansion coefficient and density respectively. V denotes the volume percentage.