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Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams
Composite Structures 98 (2013) 143–152
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Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams M. Komijani a, Y. Kiani a, S.E. Esfahani b, M.R. Eslami a,⇑ a b
Mechanical Engineering Department, Amirkabir University of Technology, Tehran 15914, Iran Mechanical Engineering Department, Islamic Azad University of South Tehran Branch, Tehran, Iran
a r t i c l e
i n f o
Article history: Available online 23 November 2012 Keywords: Functionally graded piezoelectric material Von-Karman non-linearity Thermally post-buckled state Small free vibration Timoshenko Beam Theory
a b s t r a c t The present paper deals with the small free vibration of functionally graded piezoelectric material (FGPM) beams with rectangular cross sections in pre/post-buckling regimes. The in-plane mechanical boundary conditions are considered to be immovable and various out-of-plane boundary conditions are considered. The Beam is assumed to be under in-plane thermal and electrical excitations. Each thermo-electro-mechanical property of the beam is graded across the thickness based on a power law model. The von-Karman type geometrical non-linearity is implemented to account the large deflection behavior of the beam under in-plane loadings. A Ritz-based finite element formulation is developed to discrete the motion equations. The resulted system of non-linear equations is solved via the iterative Newton–Raphson scheme. Plots of frequency in terms of loading parameter reveals the existence of bifurcation or critical states in some cases. Vibration of the beam in both pre-buckling and post-buckling states are validated with the available data in the open literature. The effects of boundary conditions, beam geometry, composition rule of constituents, actuator voltage, and thermal environment are examined through the various parametric studies. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The piezoelectric materials stand as a class of smart structures which are widely used in design problems. The ability of piezoelectric materials to surpass the vibrational motion, shape control, and delay the buckling have necessitated more investigations on the behavior of structures including piezoelectricity effects. This statement is documented by some valuable books on the subject, such as the Tzou’s one [1], or those reported by Yang [2,3]. Beam-like structures are used extensively in mechanical, civil, and structural applications. Therefore, analysis of beams made of piezoelectric materials or those incorporated with piezoelectric patch/layer(s) is of high interest. For the case when beam has a rectangular cross-shape, Yang and his co-authors [4–6] analyzed the three-dimensional behavior of electroelastic beams. In these works, extensional and transversal motions are studied. Double power-series solutions are developed in thickness and width directions. Wang and Queck [7,8] analyzed the free vibration problem of a beam integrated with piezoelectric layer(s) based on the classical beam theory. Both open and closed-circuit electrical states are examined and the effect of electrical boundary conditions on free vibration motion is investigated. Very most recently, Ke et al. [9] ⇑ Corresponding author. Tel.: +98 21 640 5844; fax: +98 21 641 9736. E-mail address: [email protected] (M.R. Eslami). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.10.047
and Ke and Wang [10] analyzed the free vibration problem of a piezoelectric beam including Eringen’s nonlocal effects in thermoelectro-mechanical field. The functionally graded materials (FGMs) are a class of novel materials in which properties of the continua varies in one or more specific direction(s). Since these structures are generally made from the mixture of a ceramic and a metal, they are able to withstand high temperature effects. Pradhan and Murmu analyzed the free vibration of FGM sandwich beam including thickness variations in thermal field [11]. Based on the first order shear deformation beam theory, Xiang and Yang [12] examined the transverse heat conduction effects on small free vibrations of symmetrically laminated FGM beams. Using an improved perturbation technique and based on a higher-order shear deformation theory, Xia and Shen [13] investigated the small and large-amplitude vibration analysis of compressively and thermally post-buckled sandwich plates with functionally graded material (FGM) face sheets in thermal environments. The results of this paper show that, as the volume fraction index increases, the fundamental frequency increases in the pre-buckling region, while in the post-buckling regime the behavior is vice versa. The free vibration analysis of an elastic rod around its post-buckled equilibrium state is addressed in the work of Neukirch et al. [14]. They employed both analytical and numerical schemes to conclude the results, before and after the buckling point.
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The FGM structures when are incorporated with piezoelectric layers are called hybrid FGMs. Vibrations of a Timoshenko beam with surface bonded piezoelectric layers in both pre/post-buckling states is studied by Li et al. [15]. In this work, shooting method is implemented to solve the post-buckling and free vibration problems of a hybrid FGM beam, clamped at both ends. Very most recently, the free vibration of a clamped hybrid FGM beam under in-plane thermal loading is investigated by Fu et al. [16]. In this work, a fully analytical method is developed to analyze the postbuckling equilibrium path and large amplitude vibrations of the beam. Due to the existence of stress concentration phenomenon in layered media, the concept of FGM structures and piezoelectric smart layers are combined together. This class of structures are called FGPMs. Such combination has the features of both piezoelectric and FGM linked together. Researches on the analysis of FGPM structures under thermoelectro mechanical loadings are limited in number. Besides, among these investigations, most of them analyze the geometrically linear response of the graded actuators. In many studies, variation of material properties in a specific direction is assumed to follow a prescribed distributed function. Huang et al. [17] developed a solution based on the two-dimensional theory of elasticity for the response of an FGPM beam with arbitrary through-the-thickness distribution of material properties. Introducing a stress function and an electrical displacement function, the equilibrium and Maxwell electrical equations are satisfied. Solution of stress function and electrical displacement function, however, are assumed to be quadratic through the span. Zhifei [18], Shi and Chen [19], and Liu and Shi [20] performed a series of investigations on orthotropic FGPM beams. In [18], Zhifei reported various analytical solutions for a cantilever beam where density varies as a cubic polynomial across the thickness. Shi and Chen in [19] consider the case of quadratic and cubic variations of elastic property and density of the beam across the thickness. With consideration of linearly graded piezoelectric parameter through-the-thickness, Liu and Shi [20] obtained the response of an FGPM beam based on the definition of stress function. Kruusing [21] obtained an analytical solution for a cantilever Euler-Bernoulli FGPM beam under the action of a shear force at the tip. When an FGPM beam is subjected to electrical or electrothermal loading, Joshi et al. [22,23] developed the bending response of the structure. It is concluded that the behavior of an FGPM beam is greatly affected by the composition rule of the constituents. Based on a layer-wise formulation, Lee [24,25] developed a finite elements method to investigate the response of the FGPM beam subjected to the combined action of thermal and electrical loadings. Yang and Xiang [26] performed a comprehensive study on the static, dynamic, and free vibration behavior of the FGPM Timoshenko beams under the action of thermal, mechanical, and electrical excitations. In this work, three mechanical equations and the Maxwell-type of the electrical equation are solved simultaneously, employing the differential quadrature (DQ) rule. Employing the classical, first order, and third order shear deformation beam theories, Komeili et al. [27] developed the finite elements and finite Fourier formulations to study the bending response of a monomorph FGPM beam under various types of loading. Dynamic response of the beam employing the Galerkin-based finite elements formulation is reported by Doroushi et al. [28] based on the third order shear deformation theory of Reddy. To the best of authors knowledge, there is no work reported on small free vibrations of a graded piezoelectric media in prebuckling and postbuckling regimes. The present paper implements the Ritz finite elements method to discrete the associated equilibrium equations. The established equations are non-linear due to the presence of von-Karman’s geometrical non-linearity in strain
components. The solution is divided into static and dynamic responses. Static response of the beam is the study of postbuckling equilibrium path under the in-plane thermoelectrical loading. The Newton–Raphson method is implemented to solve the nonlinear system of equations, iteratively. The dynamic response is the study of small free vibration in thermoelectrically pre/postbuckled states via the linear eigenvalue analysis. The variation of fundamental frequency in thermal field reveals that the behavior of the structure, depending on boundary conditions and the applied loads, may be of the bifurcation or critical point responses. Besides, in many cases the structures undergo a unique and stable equilibrium path which is free of critical states. It is important to note that, due to the manufacturing and numerical solving difficulties, in the present study only the structures with rectangular cross sections are investigated. The novel contribution of the present work is that, by applying the appropriate external voltage, the buckling phenomenon of the structure is controlled and postponed within a noticeable range. 2. Governing equations Consider a beam made of functionally graded piezoelectric materials (FGPMs) of length L, width b and thickness h. The beam is subjected to a mechanical distributed load q, temperature rise DT, and applied voltage V0 as shown in Fig. 1. It is considered that the material properties vary continuously across the thickness direction according to the power law distribution. The effective material properties P can be found as:
P ¼ PU V U þ PL V L
ð1Þ
where (PU, PL) represent the material properties at the upper and lower surfaces, respectively, and (VU, VL) are the corresponding volume fractions defined as:
VU ¼
k 1 z ; þ 2 h
VL ¼ 1 VU
ð2Þ
where k denotes the non-negative volume fraction index. In this study, the Timoshenko beam theory is used with the following displacement field
UðX; Z; tÞ ¼ U 0 ðX; tÞ þ Z WðX; tÞ WðX; Z; tÞ ¼ W 0 ðX; tÞ
ð3Þ
where (U0, W0) are the displacement components of a point on the mid-plane of the beam along axial and thickness coordinates, respectively and W stands as the rotation of the cross-section. Wang et al. [29] performed a two-dimensional elasticity solution to obtain the distribution of electrical potential across the thickness, when beam is subjected to a constant uniform mechanical load. Results of this study reveals that, for the case where a simply-supported beam is closed circuit at both top and bottom surfaces, exact distribution of electrical potential is obtained in a parabolic form, where the peak point stands at the middle. This type of distribution, also, has been used for the other types of boundary conditions, loading, and material property distribution. Also, some authors used the trigonometric functions along the thickness direction to satisfy the closed-circuit electrical conditions at the top and bottom layers [26,30,31]. In this study, considering both reverse and direct effects of a piezoelectric layer, the electric potential V is assumed to obey the following distribution [9,32]
VðX; Z; tÞ ¼ cosðbZÞUðX; tÞ þ
V0 Z h
ð4Þ
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
Fig. 1. Geometry and coordinate system of an FGPM beam.
where b ¼ ph and U(X, t) is the spatial function of the electric potential and the second term denotes the external electric voltage applied to the beam’s electrodes. The constitutive equations for the FGPM beam under thermoelectro-mechanical loading may be expressed as follows [28]:
rX ¼ Q 11 eX Q 11 a1 DT e31 EZ sXZ ¼ Q 55 cXZ e15 EX DX ¼ e15 cXZ þ k11 EX DZ ¼ e31 eX þ k33 EZ þ p3 DT
ð5Þ
ð6Þ
Since the electric field vector is the negative gradient of the total potential function, using Eq. (4), the electric field components are
EX ¼ V ;X ¼ cosðbZÞU;X EZ ¼ V ;Z ¼ b sinðbZÞU E0
1 NX ¼ A11 U 0;X þ W 20;X þ B11 W;X NTX Ae31 U þ De31 E0 2 1 M X ¼ B11 U 0;X þ W 20;X þ D11 W;X M TX Be31 U þ Ee31 E0 2 Q X ¼ K s A55 ðW 0;X þ WÞ K s De15 U;X
ð7Þ
Z h2 NTX ; MTX ¼ Q 11 a1 DTð1; ZÞdZ
where we have set E0 ¼ The governing equations may be derived on the basis of Hamiltone’s principle. According to this principle, motion equations are obtained when the following equality holds
d
ðK H þ RÞdt ¼ 0
ð8Þ
t
where the variation of the electric enthalpy dH, and the variation of the kinetic energy dK are, respectively [26]
dH ¼ b
Z
L
Z
2h
0
dK ¼ b
Z
h 2
L
Z
h 2
h2
0
ðrX deX þ K s sXZ dcXZ DX dEX DZ dEZ ÞdZ dX
qðU ;t dU ;t þ W ;t dW ;t ÞdZ dX
ð9Þ
ð10Þ
where Ks is the shear correction factor and is taken as Ks = p2/12. The virtual work dR due to the out-of-plane mechanical load q is
dR ¼ b
Z
L
qdW dX 0
ð11Þ
ð13Þ
h 2
Other quantities that are not specified, are given in Appendix A. The inertia terms are defined as
ðI0 ; I1 ; I2 Þ ¼
Z
h 2
qð1; Z; Z 2 ÞdZ
ð14Þ
2h
For the sake of generality and simplicity, the following nondimensional parameters [26] are introduced and used in the rest of this work
x¼
X ; L
U0 ¼
w ¼ W;
V0 . h
Z
ð12Þ
where NTX and MTX are thermal force and moment resultants that are defined as
where eX, cXZ, rX, sXZ, Di, and Ei represent the axial strain, shear strain, axial stress, shear stress, dielectric displacements, and the corresponding electric field components, respectively. Here, eij, kij, a1, p3 are the piezoelectric, dielectric, thermal expansion, and pyroelectric coefficients, respectively, and Q11 and Q55 are the elastic stiffness coefficients. The von-Karman type non-linear strain-displacement relations can be obtained using Eq. (3) as
1 eX ¼ U 0;X þ W 20;X þ Z W;X 2 cXZ ¼ W 0;X þ W
Using Eqs. (5)–(7), the stress resultants of the Timoshenko beam theory are
kV ¼
/¼
He33 E0 ; F e33 U0
0:5 A11 ; F e33
U ; U0 kVT
u¼
U0 ; h
w¼
W0 h
qL2 DT kq ¼ ; kT ¼ e Z A11 h F 33 U0 e T D31 E0 NX L Ee31 E0 MTX L ¼ ; kVT ¼ A11 h D11
ð15Þ
Rh where DTZ ¼ 2 h p3 DTb sinðbZÞdZ, and the quantities that are not 2 introduced are given in the Appendix A. Substituting Eqs. (6) and (7) into Eq. (8), then integrating in the thickness direction with consideration of Eq. (12), and applying the fundamental lemma of calculus, the weak-formulation of the governing equations in dimensionless form are obtained as
1 u;x þ c12 w2;x þ c13 w;x c14 / þ kVT du;x dx 2 0 Z 1 ½g11 u;tt þ g13 w;tt du dx ¼
Z
1
ð16Þ
0
1 u;x þ c12 w2;x þ c13 w;x c14 / þ kVT c12 w;x þ c22 w;x 2 0 Z 1 Z 1 c22 þ w c24 /;x dw;x dx kq dw dx ¼ ½g22 w;tt dw dx ð17Þ
Z
1
c12
0
0
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Z 1
1 c31 u;x þ c12 w2;x þ w;x c34 / þ kVT dw;x 2 0 c32 34 Þ/;x dw dx w þ ðc34 c þ c32 w;x þ ¼
1
þ
g31 u;tt þ g33 w;tt dw dx
þ
c12
ð19Þ
ð20Þ
Simply-supported ðSÞ : u ¼ w ¼ M X ¼ 0
ð21Þ
Clamped ðCÞ : u ¼ w ¼ w ¼ 0 3. Finite elements model
The Ritz-based finite element method is used to solve the weak forms of the governing equations. The variables are approximated as
j¼1
j¼1
n X
p X
j¼1
j¼1
wej ðtÞW3j ðxÞ; /ðx; tÞ ¼
ð22Þ
/ej ðtÞW4j ðxÞ
dw ¼ W2i ;
dw ¼ W3i ;
d/ ¼ W4i
ð23Þ
In this work, the quadratic interpolation functions are used to approximate the variables in the elements. Substitution of Eqs. (22) and (23) into Eqs. (16)–(19), yield the following finite element model: l m n X X X M 11 uej þ M 12 wej þ M 13 wej ij ij ij ;tt
j¼1
j¼1
;tt
p l m n X X X X e e e M 14 /ej þ K 11 K 12 K 13 ij ij uj þ ij wj þ ij wj j¼1
;tt
p X 1 e þ K 14 ij /j ¼ F i ; j¼1
;tt
j¼1
2 e K 24 ij /j ¼ F i ;
j¼1
j¼1
ði ¼ 1; . . . ; mÞ
ð25Þ
l X
e K 31 ij uj þ
j¼1 p X
;tt
j¼1
j¼1
;tt
j¼1
;tt
m n X X e e K 32 K 33 ij wj þ ij wj j¼1
j¼1
3 e K 34 ij /j ¼ F i ;
ði ¼ 1; . . . ; nÞ
ð26Þ
j¼1
j¼1
ði ¼ 1; . . . ; lÞ
p l m n X X X X M41 uej þ M 42 wej þ M 43 wej þ M 44 /ej ij ij ij ij ;tt
þ
l X j¼1
þ
p X
e K 41 ij uj þ
;tt
j¼1
j¼1
m X
n X
j¼1
j¼1
e K 42 ij wj þ
4 e K 44 ij /j ¼ F i ;
;tt
j¼1
ð24Þ
j¼1
;tt
e K 43 ij wj
ði ¼ 1; . . . ; pÞ
ð27Þ
j¼1
Definitions of the elements of Mij, Kij, and Fi are given in Appendix C. The element Eqs. (24)–(27), can be expressed in a compact form as
€ g þ ð½K L þ ½K NL1 þ ½K NL2 ÞfDg ¼ fF m g þ fF e g þ fF T g ½MfD
ð28Þ
where [M] is the matrix of inertia, and [KL], [KNL1], and [KNL2] are the linear, first order non-linear, and second order non-linear stiffness matrices, respectively, and {Fm}, {Fe}, and {FT} are the mechanical, electrical, and thermal force vectors, respectively. Besides {D} = {{u}, {/},{w}, {W}}T, is the matrix of nodal values. To study the vibration of a beam in pre/post-buckling states, the solution of the governing Eq. (28) is assumed as [33]
fDg ¼ fDs g þ fDd g
l m X X uej ðtÞW1j ðxÞ; wðx; tÞ ¼ wej ðtÞW2j ðxÞ
where Waj ðxÞ ða ¼ 1; 2; 3; 4Þ are Lagrange interpolation functions of degree (l 1), (m 1), (n 1), and (p 1), respectively. Using Eq. (22), the virtual displacements are
þ
l m n X X X e e e M24 /ej þ K 21 K 22 K 23 ij ij uj þ ij wj þ ij wj
;tt
j¼1
where the latter one is the electrical boundary condition, and the first three are the mechanical ones. It is noted that, in the solution process of this work, the natural electrical boundary condition is considered for each of the edge supports. In the stability analysis of an FGPM beam, the boundary conditions may be assumed to be immovable simply supported or clamped. Mathematical expressions for each of these edges are:
;tt
;tt
j¼1
M X ¼ 0 or w ¼ 0 c42 w;x þ c43 w þ c44 /;x ¼ 0 or / ¼ 0
j¼1
þ
þ
c12 NX w;x þ Q X ¼ 0 or w ¼ 0
du ¼ W1i ;
j¼1
p l m n X X X X M31 uej þ M 32 wej þ M 33 wej þ M 34 /ej ij ij ij ij j¼1
N X ¼ 0 or u ¼ 0
wðx; tÞ ¼
p X
;tt
j¼1
j¼1
where the constants appeared in the above equations are defined in the Appendix B. Using integration by parts in Eqs. (16)–(19), the boundary conditions become
uðx; tÞ ¼
p X j¼1
ð18Þ
1 c42 w;x þ c43 w þ c44 /;x d/;x þ c41 u;x þ c12 w2;x 2 0 c42 w;x þ / þ kT kV d/ dx ¼ 0 c43 1
;tt
j¼1
c12
Z 0
Z
l m n X X X M21 uej þ M 22 wej þ M 23 wej ij ij ij
ð29Þ
where, {Ds} is the time-independent particular solution which denotes the large displacements and is implemented to study the pre-buckling and post-buckling regimes of the beam. Besides, {Dd} is the time-dependent solution with small magnitude which is used to study the free vibration analysis of a beam in pre/post-buckling configurations. Substituting Eq. (29) into the finite element Eq. (28), results to the following sets of equations
ð½K L þ ½K NL1 þ ½K NL2 ÞfDs g ¼ fF m g þ fF e g þ fF T g
ð30Þ
€ d g þ ð½K L þ 2½K NL1 þ 3½K NL2 ÞfDd g ¼ f0g ½MfD
ð31Þ
Eq. (30) is the equation for the post-buckling analysis, and Eq. (31) is associated with the vibration analysis of the buckled structure. Due to the nonlinear effects in stiffness matrices of the above equations, an iterative method has to be used in each load step. Two commonly-used iterative schemes are the Picard iteration procedure and the Newton–Raphson method. The details of these methods are available in [34]. Both direct iteration and Newton– Raphson methods are examined to solve the non-linear finite element Eq. (30). It is important to note that, for the cases in which there exists a rapid change in the graph trend of load-deflection path, the direct iterative procedure (Picard method) does not converge for a reasonable iteration steps. This feature occurs due
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
4. Result and discussions
20
Frequency Parameter
to dependency of the solutions to the converged magnitudes of the previous load step, in each load increment. However, the Newton– Raphson method seems to be more rapid-convergent. In this study, therefore, only the Newton–Raphson method is considered to conclude the results. Using the converged magnitudes of the nodal parameters, obtained through the iterative procedure for each load step in Eq. (30), the free vibration response of the post-buckled actuator is analyzed using Eq. (31), at the updated static equilibrium position of each load increment.
15
10
Present λ=0.0 Present λ=0.2 Present λ=5.0 [15] λ=0.0 [15] λ=0.2 [15] λ =5.0
4.1. Comparison studies 5
4.2. Parametric studies To assess the non-linear pre/post-buckling free vibration behavior of an FGPM beam, a monomorph FGPM beam made of PZT-4
0 0
20
40
60
80
100
120
140
160
2
12 (L/h) α ΔT m
Fig. 2. A comparison on thermal post-buckling paths of ZrO2/Al, C C FGM beams with various volume law indices. Definition of frequency parameter is the same with Li et al. [15]. Results of Li et al. [15] are read from graph.
0.025
Present λ=0.0 Present λ=0.2 Present λ=5.0 [15] λ=0.0 [15] λ=0.2 [15] λ=5.0
0.02
(h/L) wmid
The numerical or analytical results for the free vibration of thermoelectrically post-buckled FGPM beams are not available in the literature. FGM beams are the special kinds of FGPM actuators, in which the numerical magnitudes of the electric properties are equal to zero. Thus to check the accuracy of the developed nonlinear finite element model, the results of the free vibration analysis of thermally post-buckled FGM structures in pre/post-buckling regimes, concluded using the proposed numerical method of this work are compared with some available data in the literature. Besides, the free vibration responses of FGPM actuators in stress free conditions, are compared with the results given in some other papers. Thermal post-buckling path and non-dimensional fundamental frequency versus the dimensionless temperature rise of C C FGM beams (L/h = 30) with different values of the volume fraction index in pre/post-buckling states are shown in Figs. 2 and 3. The beam is made of ZrO2/Al. Properties of the constituent are Em = 70 GPa, am = 23 106 K1, mm = 0.31Ec = 151 GPa, ac = 10 106 K1, and, mc = 0.2882. The case of k = 0 represents a beam made of ZrO2. As seen, comparisons are well justified and as expected, the fundamental frequency approaches zero at the bifurcation buckling point. It is worthwhile to notice that, at the buckling point state, the lateral stiffness value of structure is equal to zero and subsequently, the structure does not undergo any free vibration response in this situation. As the second example, in Table (1) the free vibration response of S S FGPM beams (L/h = 15, V0 = 0 V) with different values of the power law index are examined in free stress states (DT = 0). Properties of the constituents are the same with [28]. As shown in the table, the fundamental frequencies in free stress states are well-justified with those obtained based on a linear finite element analysis reported in [28]. As the third example, in Fig. (4) the non-linear free vibration analysis of a S S FGPM actuator (L/h = 15, k = 10, V0 = 0 V) is investigated. The figure represents the non-dimensional fundamental frequency versus temperature rise of the beam. Doroushi et al. [28] have used a trigonometric model (they used a complete sinusoidal model instead of the half-cosine model in this work) which is antisymmetric with respect to the mid-plane. Therefore, in this study both proposed symmetrical distribution in Eq. (4) and the antisymmetric model of [28] are examined. From the results of this figure, it is seen that the results of this study are well-justified with those reported in [28] for the free stress condition. As seen, the considered model for the distribution of electric potential in the actuators, has a significant effect on the response of the structures and may vary the parameter functions within a noticeable range.
0.015
0.01
0.005
0 0
20
40
60
80
100
120
2
12 (L/h) α ΔT m
Fig. 3. A comparison on fundamental frequency of ZrO2/Al C C FGM beams with various volume law indices. Results of Li et al. [15] are read from graph.
Table 1 Dimensionless fundamental frequencies, X of S S FGPM beams in free stress states (DT = 0) for various values of the volume fraction index (h = 0.001 m, L = 0.015 m).
Present work Doroushi et al. [28]
k=0
k=1
k = 10
0.01255 0.0126
0.01174 0.0117
0.01121 0.0112
and PZT-5H piezoelectric materials is considered. Top surface of the beam is PZT-4 rich, while the bottom one is PZT-5H rich. Table (2) represents the thermo-electro-mechanical properties of these constituents. In all the rest, thickness is assumed as h = 0.001m, unless otherwise stated. Here, the dimensionless natural frequency is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi assumed to be X ¼ xh ðq=Q 11 ÞPZT4 . In Table (3), the non-dimensional fundamental frequency for various values of uniform temperature rise parameter is given.
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M. Komijani et al. / Composite Structures 98 (2013) 143–152 −3
x 10
x 10
−3
12.8
Present [Sinusoidal distribution] Present [Cosine distribution] [28] [Sinusoidal distribution]
11.5
5 4.5 4
10.2
3.5
Ω
Ω
3 8.9
2.5 2
7.6 1.5
λ=10 λ=0.2 λ=0.01 λ=0
1
6.3
0.5 5
0 0
100
200
300
400
500
600
0
200
400
600
800
ΔT [K]
ΔT [K] Fig. 4. A comparison on fundamental frequency of a S S FGPM beam (L/h = 15, V0 = 0 V, k = 10).
Fig. 5. Effect of the volume fraction index on the pre/post buckling fundamental frequency of immovable S S FGPM beams (V0 = 0, L/h = 25).
Table 2 Thermo-electro-mechanical properties of PZT-4 and PZT-5H [28].
6
PZT-4
PZT-5H
81.3 25.6 10.0 40.3248 0.6712 108 1.0275 108 2 106 2.5 105 7500
60.6 23.0 16.604 44.9046 1.5027 108 2.554 108 10 106 0.548 105 7500
−3
5
4
Ω
P Q11 (GPa) Q55 (GPa) e31 (C m2) e15 (C m2) k11 (C2 m2 N1) k33 (C2 m2 N1) a1 (K1) p3 (C m2 K1) q (kg m3)
x 10
3
2 Table 3 The non-dimensional frequency parameter, 1000X for a C C FGPM beam with L/ h = 40 subjected to various values of DT and power law indices.
DT (K)
k=0
k = 0.2
k = 10
0 50 100 150 200 250 300 350
4.265 4.174 4.081 3.985 3.886 3.785 3.681 3.574
4.162 4.014 3.859 3.697 3.527 3.348 3.158 2.955
3.868 3.499 3.081 2.591 1.975 1.023 1.690 2.656
Three cases of power law indices are considered. As seen within the studied range of temperature parameter, the natural frequency decreases with respect to the temperature rise parameter for k = 0,0.2, while for k = 10, after a decrease up to DT = 250 K, increases from DT = 300 K. This feature may be a reason of existence of bifurcation or critical states through the loading processes, which will be discussed in detail in next figures. The curves of the fundamental frequency versus the applied temperature rise and the fundamental frequency versus the midpoint non-linear deflection of S S FGPM beams with (L/h = 25, V0 = 0 V) in pre/post-buckling regimes, are depicted in Figs. 5 and 6 respectively, for different values of the volume fraction indices. As discussed previously, buckling temperature differences (bifurcation points) are distinguishable from the fundamental
λ=10.0 λ=0.20 λ=0.01 λ=0.00
1
0
0
0.1
0.2
0.3
0.4
0.5
−w
mid
Fig. 6. The dimensionless fundamental frequency versus the mid-point dimensionless deflection of S S FGPM beam for different volume fraction indices (V0 = 0, L/h = 25).
frequency-temperature curves. Since, for the structures of the present study the buckling phenomenon occurs in the first mode of instability, in the bifurcation temperature state the fundamental frequency of the beam has to be equal to zero. It is seen that, due to the non-symmetrical distribution of material properties across the thickness, the behavior of an FGPM beam under in-plane thermal loading is not of the bifurcation-type buckling, except for the case of the reduction of an FGPM beam to a fully homogeneous one (k = 0) [35]. Apparently, volume fraction index of composition rule plays an important effect on the free vibration behavior of FGPM actuators. The associated load-deflection path for each case of positive power law index is unique and stable. In Fig. 7 the temperature rise-fundamental frequency curves of C C FGPM beams with L/h = 60, V0 = 0 V are plotted for various power law indices. Similar to the case of S S beams, when k = 0, that is the reduction of an FGPM beam to a fully homogeneous
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
2
x 10
x 10
4 3.5 3 2.5 2 1.5 1 0.5 0
0
100
200
300
400
500
ΔT [K] Fig. 8. A comparison on boundary conditions effect on the pre/post buckling fundamental frequency of an FGPM beam (V0 = 50 V, L/h = 35, k = 5).
all the other types, such behavior is not observed and the beam experiences lateral deflection in the whole range of the thermal loading procedure before the critical temperature. As expected, while in the case of bifurcation point behavior, the magnitude of the fundamental frequency in the buckling temperature is equal to zero, for the cases of critical point responses, these values are not exactly equal to zero, due to the lateral deflection induced at the onset of external electric voltage. It is seen that applying the negative voltage to the actuator electrodes increases the bifurcation/critical point temperature. This feature is valid for the constituents of this study, since the induced in-plane force in the beam may be of the compressive or tensile type, depending on the signs of piezoelectric coefficients. Consequently, considering a suitable applied external voltage, as the critical temperature increases, post-buckling strength of the beam increases too. To investigate the effect of the external voltage on critical point and the free vibration response of the FGPM actuators in the both
x 10
−3
5
λ=0.0 λ=0.2 λ=10
1.6
S−S C−S C−C
4.5 4
1.4
3.5
1.2
3
Ω Ω
S−S C−S C−C
4.5
−3
1.8
−3
5
Ω
one, the problem is posed as a bifurcation-type buckling. For the case when distribution of properties is described with a non-zero volume law index, problem is not of the bifurcation-type buckling. However, the behavior of the beam is totally different from those observed for the S S beams in Fig. 5. In the case of C C FGPM beams, thermal moments are handled by the edge supports, while due to the pyroelectric effect, at the onset of thermal loading, beam experiences the lateral deflection. The magnitude of this deflection in initial levels of loading, however, is so small but is not equal to zero. As seen in this figure, for each volume law index, there exist a unique temperature in which the magnitude of the fundamental frequency is very close to zero, and changes significantly with a little amount of temperature rise. These points may be called the critical points, since they have high importance for design purposes. However, the points cannot be considered as the bifurcation points and the non-linear behaviors of these structures are not of the primary-secondary equilibrium types. The effect of boundary conditions on the temperaturefrequency plots of the S S, C S, and C C FGPM beams (L/h = 35, V0 = 50 V, k = 5) are examined in Fig. (8). Due to the inability of simply-supported edge in supplying the extra moment, the load-frequency paths of S S and C S are completely smooth, unique, and stable. For the C C case, however, there is a critical point temperature due to the existence of an abrupt change in the associated curve. Comparing the magnitudes of the fundamental frequencies in initial levels of loading it is concluded that, as expected, the C C is the most stiff case and S S is the least one. In addition, Fig. (9) represents the fundamental frequency versus the mid-point nonlinear deflection of the proposed beam of this example for different types of the boundary conditions. Due to the existence of an external applied voltage and the nonsymmetrical distribution of the thermo-electro-mechanical properties of the constituents across the thickness direction of the beam (k = 5), it is seen that in all types of the boundary conditions (even in the C C one) the response of the structure is not of the bifurcation point behavior. The effect of the external actuator voltage on load-frequency and mid-deflection-frequency of an isotropic-homogeneous S S piezoelectric beam (L/h = 40, k = 0) is revealed in Figs. (10) and (11). As seen, in the absence of the external electric voltage, response of the beam is of the bifurcation point behavior while, in
1
2.5 2
0.8
1.5
0.6
1 0.4 0.5 0.2 0 0
0
100
200
300
400
500
600
700
ΔT [K] Fig. 7. Effect of the volume fraction index on the pre/post buckling fundamental frequency of immovable C C FGPM beams (V0 = 0, L/h = 60).
0
0.1
0.2
0.3
0.4
−wmid Fig. 9. The dimensionless fundamental frequency versus the mid-point dimensionless deflection of FGPM beam (V0 = 50 V, L/h = 35,k = 5) for different types of boundary conditions.
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M. Komijani et al. / Composite Structures 98 (2013) 143–152 −3
x 10
2.5
4
x 10
−3
3.5 2 3 2.5
Ω
Ω
1.5
2
1 1.5
V = +500V
1
0
0.5
V0 = +500V
V0 = 0.0V
V = 0.0V 0
0.5
V = −500V 0
0
V = −500V 0
0
100
200
300
400
0
500
ΔT [K]
50
x 10−4 9 8 7 6 5 4 3
V = +500V
2
150
200
250
300
350
Fig. 12. The effect of the applied actuator voltage on the pre/post-buckling fundamental frequency of a C C FGPM beam (L/h = 40, k = 10).
non-linearity are assumed to establish the inertia matrix, the non-linear stiffness matrix and force vectors. The well-known Newton–Raphson iterative scheme is used to obtain the solution of the system of non-linear equations. It is found that the behavior of unsymmetrical piezoelectric beams is totally different from the symmetrical ones. Besides, applying the suitable external voltage may alter the stability and vibration behavior of the beam within a reasonable range. The sign of the applied external voltage, however, has to be chosen according to electrical properties of the constituents. Furthermore, it is concluded that near the bifurcation points of the structures studied, the numerical values of the fundamental frequencies obtained using the free vibration analysis, approaches to zero. Therefore, small free vibrational motion of an structure may be used to detect the critical/bifurcation points of the structures.
0
V0 = 0.0V
1
Appendix A
V = −500V 0
0
100
ΔT [K]
Fig. 10. The effect of the applied actuator voltage on the fundamental frequency of S S FGPM beam (L/h = 40, k = 0) in pre/post-buckling configurations.
Ω
0
0
0.05
0.1
0.15
0.2
−w
ðA55 ; A11 ; B11 ; D11 Þ ¼
Z
pre and post-buckling states, the non-linear behavior of a C C beam (L/h = 40, k = 10) is examined in Fig. (12). Apparently, the critical point phenomenon can be postponed by applying the external voltage. For the constituents of this study, applying positive voltage to the electrodes, generates the extra axial compressive electrical force and critical temperature occurs sooner, while applying negative voltage affects vice versa. 5. Conclusion To assess the effects of direct/inverse piezoelectric effects, composition rule, and the boundary conditions on the response of a monomorph FGPM beam with rectangular cross section under the in-plane thermal/electrical loads, a Ritz-based finite elements formulation is developed. First order shear deformation theory of Timoshenko combined with the von-Karman type of geometrical
ðQ 55 ; Q 11 ; ZQ 11 ; Z 2 Q 11 ÞdZ
h=2
mid
Fig. 11. The effect of the applied actuator voltage on the dimensionless fundamental frequency versus the mid-point dimensionless deflection of S S FGPM beam (L/h = 40, k = 0). The curves associated to V0 = +500 V and V0 = 500 V are overlapped.
h=2
e A31 ; Be31 ¼
Z
h=2
e D15 ; De31 ; Ee31 ¼ e e F 11 ; F 33 ; He33 ¼
Appendix B
c12 ¼
h L
c13 ¼
B11 A11 h
c14 ¼
Ae31 U0 L A11 h
ðe31 b sinðbZÞ; Ze31 b sinðbZÞÞdZ
h=2
Z
h=2
ðe15 cosðbZÞ; e31 ; Ze31 ÞdZ h=2
Z
h=2
h=2
2 k11 cos2 ðbZÞ; k33 b2 sin ðbZÞ; k33 b sinðbZÞ dZ
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c22 ¼
K s A55 A11
K 12 ij ¼
K s A55 L A11 h
K 13 ij ¼
K s De15 U0 A11 h
K 14 ij ¼
B11 h D11
K 21 ij ¼
c32 ¼
K s A55 Lh D11
K 22 ij ¼
c33 ¼
K s A55 L2 D11
K 23 ij ¼
c23 ¼ c24 ¼ c31 ¼
c34 ¼ c34 ¼ c41 ¼ c42 ¼
Be31
K s De15
LU0
K 24 ij
D11 Be31 L
U0
Ae31 h F e33 LU0
K 32 ij ¼
De15 h K 33 ij ¼
c43 ¼ c44 ¼
F e11 F e33 L2
c43 ¼
De15 e F 33 L 0
K 34 ij ¼
U
K 41 ij ¼
U
K 42 ij ¼ 2
ðg11 ; g13 ; g22 ; g31 ; g33 Þ ¼ L
Appendix C
M 11 ij M 13 ij
¼
le
¼
¼
Z le
Z le
M 33 ij ¼ K 11 ij
Z le
M 22 ij ¼ M 31 ij
Z
¼
Z le
I0 I1 I0 I1 h I2 ; ; ; ; A11 A11 h A11 D11 D11
K 43 ij ¼ K 44 ij ¼
g11 W
1 i
1 j
W dx F 1i ¼
g13 W
1 i
3 j
W dx F 2i ¼
g22 W2i
W2j dx F 3i ¼
g31 W
3 i
g33 W3i
Z
W
le
1 i
W3j dx ;x
le
Z le
Z
dx
W3j
;x
dx
;x
c14 W1i W4j dx ;x
le
c12 w;x W2i
W1j
;x
dx
;x
Z 1 2 2 2 2 dx c12 w;x Wi Wj þ ðkVT c12 þ c22 Þ W2i W2j ;x ;x ;x ;x le 2 Z le
Z
c12 c13 w;x W2i
W3j
;x
;x
þ
c22 2 3 W W dx c12 i ;x j
F 4i ¼
;x
le
c31 W3i
Z le
W1j
;x
1 2
le
W3i
W3j
;x
;x
;x
;x
dx
;x
c31 c12 w;x W3i
Z
;x
W2j
;x
dx þ c32 W3i W2j ;x
þ
c32 3 3 W W dx c12 i j
Z 34 W3i W4j þ ðc34 c 34 ÞW3i W4j dx c ;x
le
Z
le
c41 W4i W1j
Z le
le
Z le
Z le
Z le
Z le
;x
dx
;x
;x
W2j
1 dx þ c12 c41 w;x W4i W2j ;x ;x 2
c43 W4i W3j c43 ;x
Z le
c42 W4i
Z
W dx
1 j
W
;x
1 j
c13 W1i
le
K 31 ij ¼
De15 Be31 F e33 L 0
le
Z
1 c12 w;x W1i W2j dx ;x ;x 2
Z dx ¼ c12 c14 w;x W2i W4j c24 W2i W4j
D11
F e33 L2 U0
Z
c44 W4i
;x
W4j
;x
c42 4 3 dx W W c12 i j ;x
þ W4i W4j dx
kVT W1i dx ;x
kq W2i dx kVT W3i dx ;x
ðkV kT Þ W4i dx
where le is the element length. The inertia elements which are not specified above, are equal to zero.
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
References [1] Tzou HS. Piezoelectric shells: distributed sensing and control of continua. Dordrecht: Kluwer Academic Publisher; 1993. [2] Yang JS. Introduction to the theory of piezoelectricity. New York: Springer; 2005. [3] Yang JS. The mechanics of piezoelectric structures. Singapore: World Scientific Publishing Co. Ptc. Ltd; 2006. [4] Yang JS, Zhang W. A thickness-shear high voltage piezoelectric transformer. Int J Appl Electromag Mech 1999;10(2):105–21. [5] Yang JS, Fang HY, Jiang Q. Analysis of a ceramic bimorph piezoelectric gyroscope. Int J Appl Electromag Mech 1999;10(6):459–73. [6] Yang JS. Equations for the extension and flexure of a piezoelectric beam with rectangular cross section and applications. Int J Appl Electromag Mech 1998;9(4):409–20. [7] Wang Q, Quek ST. Flexural vibration analysis of sandwich beam coupled with piezoelectric actuator. Smart Mater Struct 2000;9(1):103–9. [8] Wang Q, Quek ST. A model for the analysis of beams with embedded piezoelectric layers. J Intell Mater Syst Struct 2002;13(1):61–70. [9] Ke LL, Wang YS, Wang ZD. Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Compos Struct 2012;94(6):2038–47. [10] Ke LL, Wang YS. Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater Struct 2012;21(2) [Art. no. 025018]. [11] Pradhan SC, Murmu T. Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature. J Sound Vib 2009;321(1–2):342–62. [12] Xiang HJ, Yang J. Free and forced vibration of a laminated FGM timoshenko beam of variable thickness under heat conduction. Composites: Part B 2008;39(2):292–303. [13] Xia XK, Shen HS. Vibration of post-buckled sandwich plates with FGM face sheets in a thermal environment. J Sound Vib 2008;314(1–2):254–74. [14] Neukirch S, Frelat J, Goriely A, Maurini C. Vibrations of post-buckled rods: the singular inextensible limit. J Sound Vib 2012;331(3):704–20. [15] Li SR, SU HD, Cheng CJ. Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment. Appl Math Mech 2009;30(8):969–82. [16] Fu Y, Wang J, Mao Y. Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment. Appl Math Model 2012;36(9):4324–40. [17] Huang DJ, Ding HJ, Chen WQ. Piezoelasticity solutions for functionally graded piezoelectric beams. Smart Mater Struct 2007;16(3):687–95. [18] Zhifei S. General solution of a density functionally gradient piezoelectric cantilever and its applications. Smart Mater Struct 2002;11(1):122–9.
[19] Shi ZF, Chen Y. Functionally graded piezoelectric cantilever beam under load. Arch Appl Mech 2004;74(3–4):237–47. [20] Liu TT, Shi ZF. Bending behavior of functionally gradient piezoelectric cantilever. Ferroelectrics 2004;308(1):43–51. [21] Kruusing A. Analysis and optimization of loaded cantilever beam microactuators. Smart Mater Struct 2000;9(2):186–96. [22] Joshi S, Mukherjee A, Schmauder S. Exact solutions for characterization of electro-elastically materials. Comput Mater Sci 2003;28(3–4):548–55. [23] Joshi S, Mukherjee A, Schmauder S. Numerical characterization of functionally graded active materials under electrical and thermal fields. Smart Mater Struct 2003;12(4):571–9. [24] Lee HJ. Layerwise laminate analysis of functionally graded piezoelectric bimorph beams. J Intell Mater Syst Struct 2005;16(2):365–71. [25] Lee HJ. Layerwise analysis of thermal shape control in graded piezoelectric beams. ASME 2003 Int Mech Eng Cong Exposit 2003;68(2):79–87 [Paper no. IMECE2003-41902]. [26] Yang J, Xiang HJ. Thermo-electro-mechanical characteristics of functionally graded piezoelectric actuators. Smart Mater Struct 2007;16(3):784–97. [27] Komeili A, Akbarzadeh AH, Doroushi A, Eslami MR. Static analysis of functionally graded piezoelectric beams under thermo-electro-mechanical loads. Adv Mech Eng. 2011 [Art no. 153731]. [28] Doroushi A, Eslami MR, Komeili A. Vibration analysis and transient response of an FGPM beam under thermo-electro-mechanical loads using higher-order shear deformation theory. J Intell Mater Syst Struct 2011;22(3):231–43. [29] Wang Q, Queck ST, Sun CT, Liu X. Analysis of piezoelectric coupled circular plate. Smart Mater Struct 2001;10(2):229–39. [30] Parashar SK, Wagner UV, Hagedorn P. A modified Timoshenko beam theory for nonlinear shear-induced flexural vibrations of piezoelectric continua. Nonlinear Dyn 2004;37(3):181–205. [31] Lee PCY, Lin WS. Piezoelectrically forced vibrations of rectangular SC-cut quartz plates. J Appl Phys 1998;83(12):7822–33. [32] Liu X, Wang Q, Queck ST. Analytical solution for free vibration of piezoelectric coupled moderately thick circular plates. Int J Solids Struct 2002;39(8): 2129–51. [33] Park JS, Kim JH, Moon SH. Vibration of thermally post-buckled composite plates embedded with shape memory alloy fibers. Compos Struct 2004;63(2): 179–88. [34] Reddy JN. An introduction to nonlinear finite element analysis. Oxford, UK: Oxford University Press; 2004. [35] Komijani M, Kiani Y, Eslami MR. Non-linear thermoelectrical stability analysis of functionally graded piezoelectric material beams. J Intell Mater Syst Struct. 2012. http://dx.doi.org/10.1177/1045389x12461079.
Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams M. Komijani a, Y. Kiani a, S.E. Esfahani b, M.R. Eslami a,⇑ a b
Mechanical Engineering Department, Amirkabir University of Technology, Tehran 15914, Iran Mechanical Engineering Department, Islamic Azad University of South Tehran Branch, Tehran, Iran
a r t i c l e
i n f o
Article history: Available online 23 November 2012 Keywords: Functionally graded piezoelectric material Von-Karman non-linearity Thermally post-buckled state Small free vibration Timoshenko Beam Theory
a b s t r a c t The present paper deals with the small free vibration of functionally graded piezoelectric material (FGPM) beams with rectangular cross sections in pre/post-buckling regimes. The in-plane mechanical boundary conditions are considered to be immovable and various out-of-plane boundary conditions are considered. The Beam is assumed to be under in-plane thermal and electrical excitations. Each thermo-electro-mechanical property of the beam is graded across the thickness based on a power law model. The von-Karman type geometrical non-linearity is implemented to account the large deflection behavior of the beam under in-plane loadings. A Ritz-based finite element formulation is developed to discrete the motion equations. The resulted system of non-linear equations is solved via the iterative Newton–Raphson scheme. Plots of frequency in terms of loading parameter reveals the existence of bifurcation or critical states in some cases. Vibration of the beam in both pre-buckling and post-buckling states are validated with the available data in the open literature. The effects of boundary conditions, beam geometry, composition rule of constituents, actuator voltage, and thermal environment are examined through the various parametric studies. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The piezoelectric materials stand as a class of smart structures which are widely used in design problems. The ability of piezoelectric materials to surpass the vibrational motion, shape control, and delay the buckling have necessitated more investigations on the behavior of structures including piezoelectricity effects. This statement is documented by some valuable books on the subject, such as the Tzou’s one [1], or those reported by Yang [2,3]. Beam-like structures are used extensively in mechanical, civil, and structural applications. Therefore, analysis of beams made of piezoelectric materials or those incorporated with piezoelectric patch/layer(s) is of high interest. For the case when beam has a rectangular cross-shape, Yang and his co-authors [4–6] analyzed the three-dimensional behavior of electroelastic beams. In these works, extensional and transversal motions are studied. Double power-series solutions are developed in thickness and width directions. Wang and Queck [7,8] analyzed the free vibration problem of a beam integrated with piezoelectric layer(s) based on the classical beam theory. Both open and closed-circuit electrical states are examined and the effect of electrical boundary conditions on free vibration motion is investigated. Very most recently, Ke et al. [9] ⇑ Corresponding author. Tel.: +98 21 640 5844; fax: +98 21 641 9736. E-mail address: [email protected] (M.R. Eslami). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.10.047
and Ke and Wang [10] analyzed the free vibration problem of a piezoelectric beam including Eringen’s nonlocal effects in thermoelectro-mechanical field. The functionally graded materials (FGMs) are a class of novel materials in which properties of the continua varies in one or more specific direction(s). Since these structures are generally made from the mixture of a ceramic and a metal, they are able to withstand high temperature effects. Pradhan and Murmu analyzed the free vibration of FGM sandwich beam including thickness variations in thermal field [11]. Based on the first order shear deformation beam theory, Xiang and Yang [12] examined the transverse heat conduction effects on small free vibrations of symmetrically laminated FGM beams. Using an improved perturbation technique and based on a higher-order shear deformation theory, Xia and Shen [13] investigated the small and large-amplitude vibration analysis of compressively and thermally post-buckled sandwich plates with functionally graded material (FGM) face sheets in thermal environments. The results of this paper show that, as the volume fraction index increases, the fundamental frequency increases in the pre-buckling region, while in the post-buckling regime the behavior is vice versa. The free vibration analysis of an elastic rod around its post-buckled equilibrium state is addressed in the work of Neukirch et al. [14]. They employed both analytical and numerical schemes to conclude the results, before and after the buckling point.
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
The FGM structures when are incorporated with piezoelectric layers are called hybrid FGMs. Vibrations of a Timoshenko beam with surface bonded piezoelectric layers in both pre/post-buckling states is studied by Li et al. [15]. In this work, shooting method is implemented to solve the post-buckling and free vibration problems of a hybrid FGM beam, clamped at both ends. Very most recently, the free vibration of a clamped hybrid FGM beam under in-plane thermal loading is investigated by Fu et al. [16]. In this work, a fully analytical method is developed to analyze the postbuckling equilibrium path and large amplitude vibrations of the beam. Due to the existence of stress concentration phenomenon in layered media, the concept of FGM structures and piezoelectric smart layers are combined together. This class of structures are called FGPMs. Such combination has the features of both piezoelectric and FGM linked together. Researches on the analysis of FGPM structures under thermoelectro mechanical loadings are limited in number. Besides, among these investigations, most of them analyze the geometrically linear response of the graded actuators. In many studies, variation of material properties in a specific direction is assumed to follow a prescribed distributed function. Huang et al. [17] developed a solution based on the two-dimensional theory of elasticity for the response of an FGPM beam with arbitrary through-the-thickness distribution of material properties. Introducing a stress function and an electrical displacement function, the equilibrium and Maxwell electrical equations are satisfied. Solution of stress function and electrical displacement function, however, are assumed to be quadratic through the span. Zhifei [18], Shi and Chen [19], and Liu and Shi [20] performed a series of investigations on orthotropic FGPM beams. In [18], Zhifei reported various analytical solutions for a cantilever beam where density varies as a cubic polynomial across the thickness. Shi and Chen in [19] consider the case of quadratic and cubic variations of elastic property and density of the beam across the thickness. With consideration of linearly graded piezoelectric parameter through-the-thickness, Liu and Shi [20] obtained the response of an FGPM beam based on the definition of stress function. Kruusing [21] obtained an analytical solution for a cantilever Euler-Bernoulli FGPM beam under the action of a shear force at the tip. When an FGPM beam is subjected to electrical or electrothermal loading, Joshi et al. [22,23] developed the bending response of the structure. It is concluded that the behavior of an FGPM beam is greatly affected by the composition rule of the constituents. Based on a layer-wise formulation, Lee [24,25] developed a finite elements method to investigate the response of the FGPM beam subjected to the combined action of thermal and electrical loadings. Yang and Xiang [26] performed a comprehensive study on the static, dynamic, and free vibration behavior of the FGPM Timoshenko beams under the action of thermal, mechanical, and electrical excitations. In this work, three mechanical equations and the Maxwell-type of the electrical equation are solved simultaneously, employing the differential quadrature (DQ) rule. Employing the classical, first order, and third order shear deformation beam theories, Komeili et al. [27] developed the finite elements and finite Fourier formulations to study the bending response of a monomorph FGPM beam under various types of loading. Dynamic response of the beam employing the Galerkin-based finite elements formulation is reported by Doroushi et al. [28] based on the third order shear deformation theory of Reddy. To the best of authors knowledge, there is no work reported on small free vibrations of a graded piezoelectric media in prebuckling and postbuckling regimes. The present paper implements the Ritz finite elements method to discrete the associated equilibrium equations. The established equations are non-linear due to the presence of von-Karman’s geometrical non-linearity in strain
components. The solution is divided into static and dynamic responses. Static response of the beam is the study of postbuckling equilibrium path under the in-plane thermoelectrical loading. The Newton–Raphson method is implemented to solve the nonlinear system of equations, iteratively. The dynamic response is the study of small free vibration in thermoelectrically pre/postbuckled states via the linear eigenvalue analysis. The variation of fundamental frequency in thermal field reveals that the behavior of the structure, depending on boundary conditions and the applied loads, may be of the bifurcation or critical point responses. Besides, in many cases the structures undergo a unique and stable equilibrium path which is free of critical states. It is important to note that, due to the manufacturing and numerical solving difficulties, in the present study only the structures with rectangular cross sections are investigated. The novel contribution of the present work is that, by applying the appropriate external voltage, the buckling phenomenon of the structure is controlled and postponed within a noticeable range. 2. Governing equations Consider a beam made of functionally graded piezoelectric materials (FGPMs) of length L, width b and thickness h. The beam is subjected to a mechanical distributed load q, temperature rise DT, and applied voltage V0 as shown in Fig. 1. It is considered that the material properties vary continuously across the thickness direction according to the power law distribution. The effective material properties P can be found as:
P ¼ PU V U þ PL V L
ð1Þ
where (PU, PL) represent the material properties at the upper and lower surfaces, respectively, and (VU, VL) are the corresponding volume fractions defined as:
VU ¼
k 1 z ; þ 2 h
VL ¼ 1 VU
ð2Þ
where k denotes the non-negative volume fraction index. In this study, the Timoshenko beam theory is used with the following displacement field
UðX; Z; tÞ ¼ U 0 ðX; tÞ þ Z WðX; tÞ WðX; Z; tÞ ¼ W 0 ðX; tÞ
ð3Þ
where (U0, W0) are the displacement components of a point on the mid-plane of the beam along axial and thickness coordinates, respectively and W stands as the rotation of the cross-section. Wang et al. [29] performed a two-dimensional elasticity solution to obtain the distribution of electrical potential across the thickness, when beam is subjected to a constant uniform mechanical load. Results of this study reveals that, for the case where a simply-supported beam is closed circuit at both top and bottom surfaces, exact distribution of electrical potential is obtained in a parabolic form, where the peak point stands at the middle. This type of distribution, also, has been used for the other types of boundary conditions, loading, and material property distribution. Also, some authors used the trigonometric functions along the thickness direction to satisfy the closed-circuit electrical conditions at the top and bottom layers [26,30,31]. In this study, considering both reverse and direct effects of a piezoelectric layer, the electric potential V is assumed to obey the following distribution [9,32]
VðX; Z; tÞ ¼ cosðbZÞUðX; tÞ þ
V0 Z h
ð4Þ
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
Fig. 1. Geometry and coordinate system of an FGPM beam.
where b ¼ ph and U(X, t) is the spatial function of the electric potential and the second term denotes the external electric voltage applied to the beam’s electrodes. The constitutive equations for the FGPM beam under thermoelectro-mechanical loading may be expressed as follows [28]:
rX ¼ Q 11 eX Q 11 a1 DT e31 EZ sXZ ¼ Q 55 cXZ e15 EX DX ¼ e15 cXZ þ k11 EX DZ ¼ e31 eX þ k33 EZ þ p3 DT
ð5Þ
ð6Þ
Since the electric field vector is the negative gradient of the total potential function, using Eq. (4), the electric field components are
EX ¼ V ;X ¼ cosðbZÞU;X EZ ¼ V ;Z ¼ b sinðbZÞU E0
1 NX ¼ A11 U 0;X þ W 20;X þ B11 W;X NTX Ae31 U þ De31 E0 2 1 M X ¼ B11 U 0;X þ W 20;X þ D11 W;X M TX Be31 U þ Ee31 E0 2 Q X ¼ K s A55 ðW 0;X þ WÞ K s De15 U;X
ð7Þ
Z h2 NTX ; MTX ¼ Q 11 a1 DTð1; ZÞdZ
where we have set E0 ¼ The governing equations may be derived on the basis of Hamiltone’s principle. According to this principle, motion equations are obtained when the following equality holds
d
ðK H þ RÞdt ¼ 0
ð8Þ
t
where the variation of the electric enthalpy dH, and the variation of the kinetic energy dK are, respectively [26]
dH ¼ b
Z
L
Z
2h
0
dK ¼ b
Z
h 2
L
Z
h 2
h2
0
ðrX deX þ K s sXZ dcXZ DX dEX DZ dEZ ÞdZ dX
qðU ;t dU ;t þ W ;t dW ;t ÞdZ dX
ð9Þ
ð10Þ
where Ks is the shear correction factor and is taken as Ks = p2/12. The virtual work dR due to the out-of-plane mechanical load q is
dR ¼ b
Z
L
qdW dX 0
ð11Þ
ð13Þ
h 2
Other quantities that are not specified, are given in Appendix A. The inertia terms are defined as
ðI0 ; I1 ; I2 Þ ¼
Z
h 2
qð1; Z; Z 2 ÞdZ
ð14Þ
2h
For the sake of generality and simplicity, the following nondimensional parameters [26] are introduced and used in the rest of this work
x¼
X ; L
U0 ¼
w ¼ W;
V0 . h
Z
ð12Þ
where NTX and MTX are thermal force and moment resultants that are defined as
where eX, cXZ, rX, sXZ, Di, and Ei represent the axial strain, shear strain, axial stress, shear stress, dielectric displacements, and the corresponding electric field components, respectively. Here, eij, kij, a1, p3 are the piezoelectric, dielectric, thermal expansion, and pyroelectric coefficients, respectively, and Q11 and Q55 are the elastic stiffness coefficients. The von-Karman type non-linear strain-displacement relations can be obtained using Eq. (3) as
1 eX ¼ U 0;X þ W 20;X þ Z W;X 2 cXZ ¼ W 0;X þ W
Using Eqs. (5)–(7), the stress resultants of the Timoshenko beam theory are
kV ¼
/¼
He33 E0 ; F e33 U0
0:5 A11 ; F e33
U ; U0 kVT
u¼
U0 ; h
w¼
W0 h
qL2 DT kq ¼ ; kT ¼ e Z A11 h F 33 U0 e T D31 E0 NX L Ee31 E0 MTX L ¼ ; kVT ¼ A11 h D11
ð15Þ
Rh where DTZ ¼ 2 h p3 DTb sinðbZÞdZ, and the quantities that are not 2 introduced are given in the Appendix A. Substituting Eqs. (6) and (7) into Eq. (8), then integrating in the thickness direction with consideration of Eq. (12), and applying the fundamental lemma of calculus, the weak-formulation of the governing equations in dimensionless form are obtained as
1 u;x þ c12 w2;x þ c13 w;x c14 / þ kVT du;x dx 2 0 Z 1 ½g11 u;tt þ g13 w;tt du dx ¼
Z
1
ð16Þ
0
1 u;x þ c12 w2;x þ c13 w;x c14 / þ kVT c12 w;x þ c22 w;x 2 0 Z 1 Z 1 c22 þ w c24 /;x dw;x dx kq dw dx ¼ ½g22 w;tt dw dx ð17Þ
Z
1
c12
0
0
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
Z 1
1 c31 u;x þ c12 w2;x þ w;x c34 / þ kVT dw;x 2 0 c32 34 Þ/;x dw dx w þ ðc34 c þ c32 w;x þ ¼
1
þ
g31 u;tt þ g33 w;tt dw dx
þ
c12
ð19Þ
ð20Þ
Simply-supported ðSÞ : u ¼ w ¼ M X ¼ 0
ð21Þ
Clamped ðCÞ : u ¼ w ¼ w ¼ 0 3. Finite elements model
The Ritz-based finite element method is used to solve the weak forms of the governing equations. The variables are approximated as
j¼1
j¼1
n X
p X
j¼1
j¼1
wej ðtÞW3j ðxÞ; /ðx; tÞ ¼
ð22Þ
/ej ðtÞW4j ðxÞ
dw ¼ W2i ;
dw ¼ W3i ;
d/ ¼ W4i
ð23Þ
In this work, the quadratic interpolation functions are used to approximate the variables in the elements. Substitution of Eqs. (22) and (23) into Eqs. (16)–(19), yield the following finite element model: l m n X X X M 11 uej þ M 12 wej þ M 13 wej ij ij ij ;tt
j¼1
j¼1
;tt
p l m n X X X X e e e M 14 /ej þ K 11 K 12 K 13 ij ij uj þ ij wj þ ij wj j¼1
;tt
p X 1 e þ K 14 ij /j ¼ F i ; j¼1
;tt
j¼1
2 e K 24 ij /j ¼ F i ;
j¼1
j¼1
ði ¼ 1; . . . ; mÞ
ð25Þ
l X
e K 31 ij uj þ
j¼1 p X
;tt
j¼1
j¼1
;tt
j¼1
;tt
m n X X e e K 32 K 33 ij wj þ ij wj j¼1
j¼1
3 e K 34 ij /j ¼ F i ;
ði ¼ 1; . . . ; nÞ
ð26Þ
j¼1
j¼1
ði ¼ 1; . . . ; lÞ
p l m n X X X X M41 uej þ M 42 wej þ M 43 wej þ M 44 /ej ij ij ij ij ;tt
þ
l X j¼1
þ
p X
e K 41 ij uj þ
;tt
j¼1
j¼1
m X
n X
j¼1
j¼1
e K 42 ij wj þ
4 e K 44 ij /j ¼ F i ;
;tt
j¼1
ð24Þ
j¼1
;tt
e K 43 ij wj
ði ¼ 1; . . . ; pÞ
ð27Þ
j¼1
Definitions of the elements of Mij, Kij, and Fi are given in Appendix C. The element Eqs. (24)–(27), can be expressed in a compact form as
€ g þ ð½K L þ ½K NL1 þ ½K NL2 ÞfDg ¼ fF m g þ fF e g þ fF T g ½MfD
ð28Þ
where [M] is the matrix of inertia, and [KL], [KNL1], and [KNL2] are the linear, first order non-linear, and second order non-linear stiffness matrices, respectively, and {Fm}, {Fe}, and {FT} are the mechanical, electrical, and thermal force vectors, respectively. Besides {D} = {{u}, {/},{w}, {W}}T, is the matrix of nodal values. To study the vibration of a beam in pre/post-buckling states, the solution of the governing Eq. (28) is assumed as [33]
fDg ¼ fDs g þ fDd g
l m X X uej ðtÞW1j ðxÞ; wðx; tÞ ¼ wej ðtÞW2j ðxÞ
where Waj ðxÞ ða ¼ 1; 2; 3; 4Þ are Lagrange interpolation functions of degree (l 1), (m 1), (n 1), and (p 1), respectively. Using Eq. (22), the virtual displacements are
þ
l m n X X X e e e M24 /ej þ K 21 K 22 K 23 ij ij uj þ ij wj þ ij wj
;tt
j¼1
where the latter one is the electrical boundary condition, and the first three are the mechanical ones. It is noted that, in the solution process of this work, the natural electrical boundary condition is considered for each of the edge supports. In the stability analysis of an FGPM beam, the boundary conditions may be assumed to be immovable simply supported or clamped. Mathematical expressions for each of these edges are:
;tt
;tt
j¼1
M X ¼ 0 or w ¼ 0 c42 w;x þ c43 w þ c44 /;x ¼ 0 or / ¼ 0
j¼1
þ
þ
c12 NX w;x þ Q X ¼ 0 or w ¼ 0
du ¼ W1i ;
j¼1
p l m n X X X X M31 uej þ M 32 wej þ M 33 wej þ M 34 /ej ij ij ij ij j¼1
N X ¼ 0 or u ¼ 0
wðx; tÞ ¼
p X
;tt
j¼1
j¼1
where the constants appeared in the above equations are defined in the Appendix B. Using integration by parts in Eqs. (16)–(19), the boundary conditions become
uðx; tÞ ¼
p X j¼1
ð18Þ
1 c42 w;x þ c43 w þ c44 /;x d/;x þ c41 u;x þ c12 w2;x 2 0 c42 w;x þ / þ kT kV d/ dx ¼ 0 c43 1
;tt
j¼1
c12
Z 0
Z
l m n X X X M21 uej þ M 22 wej þ M 23 wej ij ij ij
ð29Þ
where, {Ds} is the time-independent particular solution which denotes the large displacements and is implemented to study the pre-buckling and post-buckling regimes of the beam. Besides, {Dd} is the time-dependent solution with small magnitude which is used to study the free vibration analysis of a beam in pre/post-buckling configurations. Substituting Eq. (29) into the finite element Eq. (28), results to the following sets of equations
ð½K L þ ½K NL1 þ ½K NL2 ÞfDs g ¼ fF m g þ fF e g þ fF T g
ð30Þ
€ d g þ ð½K L þ 2½K NL1 þ 3½K NL2 ÞfDd g ¼ f0g ½MfD
ð31Þ
Eq. (30) is the equation for the post-buckling analysis, and Eq. (31) is associated with the vibration analysis of the buckled structure. Due to the nonlinear effects in stiffness matrices of the above equations, an iterative method has to be used in each load step. Two commonly-used iterative schemes are the Picard iteration procedure and the Newton–Raphson method. The details of these methods are available in [34]. Both direct iteration and Newton– Raphson methods are examined to solve the non-linear finite element Eq. (30). It is important to note that, for the cases in which there exists a rapid change in the graph trend of load-deflection path, the direct iterative procedure (Picard method) does not converge for a reasonable iteration steps. This feature occurs due
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4. Result and discussions
20
Frequency Parameter
to dependency of the solutions to the converged magnitudes of the previous load step, in each load increment. However, the Newton– Raphson method seems to be more rapid-convergent. In this study, therefore, only the Newton–Raphson method is considered to conclude the results. Using the converged magnitudes of the nodal parameters, obtained through the iterative procedure for each load step in Eq. (30), the free vibration response of the post-buckled actuator is analyzed using Eq. (31), at the updated static equilibrium position of each load increment.
15
10
Present λ=0.0 Present λ=0.2 Present λ=5.0 [15] λ=0.0 [15] λ=0.2 [15] λ =5.0
4.1. Comparison studies 5
4.2. Parametric studies To assess the non-linear pre/post-buckling free vibration behavior of an FGPM beam, a monomorph FGPM beam made of PZT-4
0 0
20
40
60
80
100
120
140
160
2
12 (L/h) α ΔT m
Fig. 2. A comparison on thermal post-buckling paths of ZrO2/Al, C C FGM beams with various volume law indices. Definition of frequency parameter is the same with Li et al. [15]. Results of Li et al. [15] are read from graph.
0.025
Present λ=0.0 Present λ=0.2 Present λ=5.0 [15] λ=0.0 [15] λ=0.2 [15] λ=5.0
0.02
(h/L) wmid
The numerical or analytical results for the free vibration of thermoelectrically post-buckled FGPM beams are not available in the literature. FGM beams are the special kinds of FGPM actuators, in which the numerical magnitudes of the electric properties are equal to zero. Thus to check the accuracy of the developed nonlinear finite element model, the results of the free vibration analysis of thermally post-buckled FGM structures in pre/post-buckling regimes, concluded using the proposed numerical method of this work are compared with some available data in the literature. Besides, the free vibration responses of FGPM actuators in stress free conditions, are compared with the results given in some other papers. Thermal post-buckling path and non-dimensional fundamental frequency versus the dimensionless temperature rise of C C FGM beams (L/h = 30) with different values of the volume fraction index in pre/post-buckling states are shown in Figs. 2 and 3. The beam is made of ZrO2/Al. Properties of the constituent are Em = 70 GPa, am = 23 106 K1, mm = 0.31Ec = 151 GPa, ac = 10 106 K1, and, mc = 0.2882. The case of k = 0 represents a beam made of ZrO2. As seen, comparisons are well justified and as expected, the fundamental frequency approaches zero at the bifurcation buckling point. It is worthwhile to notice that, at the buckling point state, the lateral stiffness value of structure is equal to zero and subsequently, the structure does not undergo any free vibration response in this situation. As the second example, in Table (1) the free vibration response of S S FGPM beams (L/h = 15, V0 = 0 V) with different values of the power law index are examined in free stress states (DT = 0). Properties of the constituents are the same with [28]. As shown in the table, the fundamental frequencies in free stress states are well-justified with those obtained based on a linear finite element analysis reported in [28]. As the third example, in Fig. (4) the non-linear free vibration analysis of a S S FGPM actuator (L/h = 15, k = 10, V0 = 0 V) is investigated. The figure represents the non-dimensional fundamental frequency versus temperature rise of the beam. Doroushi et al. [28] have used a trigonometric model (they used a complete sinusoidal model instead of the half-cosine model in this work) which is antisymmetric with respect to the mid-plane. Therefore, in this study both proposed symmetrical distribution in Eq. (4) and the antisymmetric model of [28] are examined. From the results of this figure, it is seen that the results of this study are well-justified with those reported in [28] for the free stress condition. As seen, the considered model for the distribution of electric potential in the actuators, has a significant effect on the response of the structures and may vary the parameter functions within a noticeable range.
0.015
0.01
0.005
0 0
20
40
60
80
100
120
2
12 (L/h) α ΔT m
Fig. 3. A comparison on fundamental frequency of ZrO2/Al C C FGM beams with various volume law indices. Results of Li et al. [15] are read from graph.
Table 1 Dimensionless fundamental frequencies, X of S S FGPM beams in free stress states (DT = 0) for various values of the volume fraction index (h = 0.001 m, L = 0.015 m).
Present work Doroushi et al. [28]
k=0
k=1
k = 10
0.01255 0.0126
0.01174 0.0117
0.01121 0.0112
and PZT-5H piezoelectric materials is considered. Top surface of the beam is PZT-4 rich, while the bottom one is PZT-5H rich. Table (2) represents the thermo-electro-mechanical properties of these constituents. In all the rest, thickness is assumed as h = 0.001m, unless otherwise stated. Here, the dimensionless natural frequency is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi assumed to be X ¼ xh ðq=Q 11 ÞPZT4 . In Table (3), the non-dimensional fundamental frequency for various values of uniform temperature rise parameter is given.
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M. Komijani et al. / Composite Structures 98 (2013) 143–152 −3
x 10
x 10
−3
12.8
Present [Sinusoidal distribution] Present [Cosine distribution] [28] [Sinusoidal distribution]
11.5
5 4.5 4
10.2
3.5
Ω
Ω
3 8.9
2.5 2
7.6 1.5
λ=10 λ=0.2 λ=0.01 λ=0
1
6.3
0.5 5
0 0
100
200
300
400
500
600
0
200
400
600
800
ΔT [K]
ΔT [K] Fig. 4. A comparison on fundamental frequency of a S S FGPM beam (L/h = 15, V0 = 0 V, k = 10).
Fig. 5. Effect of the volume fraction index on the pre/post buckling fundamental frequency of immovable S S FGPM beams (V0 = 0, L/h = 25).
Table 2 Thermo-electro-mechanical properties of PZT-4 and PZT-5H [28].
6
PZT-4
PZT-5H
81.3 25.6 10.0 40.3248 0.6712 108 1.0275 108 2 106 2.5 105 7500
60.6 23.0 16.604 44.9046 1.5027 108 2.554 108 10 106 0.548 105 7500
−3
5
4
Ω
P Q11 (GPa) Q55 (GPa) e31 (C m2) e15 (C m2) k11 (C2 m2 N1) k33 (C2 m2 N1) a1 (K1) p3 (C m2 K1) q (kg m3)
x 10
3
2 Table 3 The non-dimensional frequency parameter, 1000X for a C C FGPM beam with L/ h = 40 subjected to various values of DT and power law indices.
DT (K)
k=0
k = 0.2
k = 10
0 50 100 150 200 250 300 350
4.265 4.174 4.081 3.985 3.886 3.785 3.681 3.574
4.162 4.014 3.859 3.697 3.527 3.348 3.158 2.955
3.868 3.499 3.081 2.591 1.975 1.023 1.690 2.656
Three cases of power law indices are considered. As seen within the studied range of temperature parameter, the natural frequency decreases with respect to the temperature rise parameter for k = 0,0.2, while for k = 10, after a decrease up to DT = 250 K, increases from DT = 300 K. This feature may be a reason of existence of bifurcation or critical states through the loading processes, which will be discussed in detail in next figures. The curves of the fundamental frequency versus the applied temperature rise and the fundamental frequency versus the midpoint non-linear deflection of S S FGPM beams with (L/h = 25, V0 = 0 V) in pre/post-buckling regimes, are depicted in Figs. 5 and 6 respectively, for different values of the volume fraction indices. As discussed previously, buckling temperature differences (bifurcation points) are distinguishable from the fundamental
λ=10.0 λ=0.20 λ=0.01 λ=0.00
1
0
0
0.1
0.2
0.3
0.4
0.5
−w
mid
Fig. 6. The dimensionless fundamental frequency versus the mid-point dimensionless deflection of S S FGPM beam for different volume fraction indices (V0 = 0, L/h = 25).
frequency-temperature curves. Since, for the structures of the present study the buckling phenomenon occurs in the first mode of instability, in the bifurcation temperature state the fundamental frequency of the beam has to be equal to zero. It is seen that, due to the non-symmetrical distribution of material properties across the thickness, the behavior of an FGPM beam under in-plane thermal loading is not of the bifurcation-type buckling, except for the case of the reduction of an FGPM beam to a fully homogeneous one (k = 0) [35]. Apparently, volume fraction index of composition rule plays an important effect on the free vibration behavior of FGPM actuators. The associated load-deflection path for each case of positive power law index is unique and stable. In Fig. 7 the temperature rise-fundamental frequency curves of C C FGPM beams with L/h = 60, V0 = 0 V are plotted for various power law indices. Similar to the case of S S beams, when k = 0, that is the reduction of an FGPM beam to a fully homogeneous
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2
x 10
x 10
4 3.5 3 2.5 2 1.5 1 0.5 0
0
100
200
300
400
500
ΔT [K] Fig. 8. A comparison on boundary conditions effect on the pre/post buckling fundamental frequency of an FGPM beam (V0 = 50 V, L/h = 35, k = 5).
all the other types, such behavior is not observed and the beam experiences lateral deflection in the whole range of the thermal loading procedure before the critical temperature. As expected, while in the case of bifurcation point behavior, the magnitude of the fundamental frequency in the buckling temperature is equal to zero, for the cases of critical point responses, these values are not exactly equal to zero, due to the lateral deflection induced at the onset of external electric voltage. It is seen that applying the negative voltage to the actuator electrodes increases the bifurcation/critical point temperature. This feature is valid for the constituents of this study, since the induced in-plane force in the beam may be of the compressive or tensile type, depending on the signs of piezoelectric coefficients. Consequently, considering a suitable applied external voltage, as the critical temperature increases, post-buckling strength of the beam increases too. To investigate the effect of the external voltage on critical point and the free vibration response of the FGPM actuators in the both
x 10
−3
5
λ=0.0 λ=0.2 λ=10
1.6
S−S C−S C−C
4.5 4
1.4
3.5
1.2
3
Ω Ω
S−S C−S C−C
4.5
−3
1.8
−3
5
Ω
one, the problem is posed as a bifurcation-type buckling. For the case when distribution of properties is described with a non-zero volume law index, problem is not of the bifurcation-type buckling. However, the behavior of the beam is totally different from those observed for the S S beams in Fig. 5. In the case of C C FGPM beams, thermal moments are handled by the edge supports, while due to the pyroelectric effect, at the onset of thermal loading, beam experiences the lateral deflection. The magnitude of this deflection in initial levels of loading, however, is so small but is not equal to zero. As seen in this figure, for each volume law index, there exist a unique temperature in which the magnitude of the fundamental frequency is very close to zero, and changes significantly with a little amount of temperature rise. These points may be called the critical points, since they have high importance for design purposes. However, the points cannot be considered as the bifurcation points and the non-linear behaviors of these structures are not of the primary-secondary equilibrium types. The effect of boundary conditions on the temperaturefrequency plots of the S S, C S, and C C FGPM beams (L/h = 35, V0 = 50 V, k = 5) are examined in Fig. (8). Due to the inability of simply-supported edge in supplying the extra moment, the load-frequency paths of S S and C S are completely smooth, unique, and stable. For the C C case, however, there is a critical point temperature due to the existence of an abrupt change in the associated curve. Comparing the magnitudes of the fundamental frequencies in initial levels of loading it is concluded that, as expected, the C C is the most stiff case and S S is the least one. In addition, Fig. (9) represents the fundamental frequency versus the mid-point nonlinear deflection of the proposed beam of this example for different types of the boundary conditions. Due to the existence of an external applied voltage and the nonsymmetrical distribution of the thermo-electro-mechanical properties of the constituents across the thickness direction of the beam (k = 5), it is seen that in all types of the boundary conditions (even in the C C one) the response of the structure is not of the bifurcation point behavior. The effect of the external actuator voltage on load-frequency and mid-deflection-frequency of an isotropic-homogeneous S S piezoelectric beam (L/h = 40, k = 0) is revealed in Figs. (10) and (11). As seen, in the absence of the external electric voltage, response of the beam is of the bifurcation point behavior while, in
1
2.5 2
0.8
1.5
0.6
1 0.4 0.5 0.2 0 0
0
100
200
300
400
500
600
700
ΔT [K] Fig. 7. Effect of the volume fraction index on the pre/post buckling fundamental frequency of immovable C C FGPM beams (V0 = 0, L/h = 60).
0
0.1
0.2
0.3
0.4
−wmid Fig. 9. The dimensionless fundamental frequency versus the mid-point dimensionless deflection of FGPM beam (V0 = 50 V, L/h = 35,k = 5) for different types of boundary conditions.
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x 10
2.5
4
x 10
−3
3.5 2 3 2.5
Ω
Ω
1.5
2
1 1.5
V = +500V
1
0
0.5
V0 = +500V
V0 = 0.0V
V = 0.0V 0
0.5
V = −500V 0
0
V = −500V 0
0
100
200
300
400
0
500
ΔT [K]
50
x 10−4 9 8 7 6 5 4 3
V = +500V
2
150
200
250
300
350
Fig. 12. The effect of the applied actuator voltage on the pre/post-buckling fundamental frequency of a C C FGPM beam (L/h = 40, k = 10).
non-linearity are assumed to establish the inertia matrix, the non-linear stiffness matrix and force vectors. The well-known Newton–Raphson iterative scheme is used to obtain the solution of the system of non-linear equations. It is found that the behavior of unsymmetrical piezoelectric beams is totally different from the symmetrical ones. Besides, applying the suitable external voltage may alter the stability and vibration behavior of the beam within a reasonable range. The sign of the applied external voltage, however, has to be chosen according to electrical properties of the constituents. Furthermore, it is concluded that near the bifurcation points of the structures studied, the numerical values of the fundamental frequencies obtained using the free vibration analysis, approaches to zero. Therefore, small free vibrational motion of an structure may be used to detect the critical/bifurcation points of the structures.
0
V0 = 0.0V
1
Appendix A
V = −500V 0
0
100
ΔT [K]
Fig. 10. The effect of the applied actuator voltage on the fundamental frequency of S S FGPM beam (L/h = 40, k = 0) in pre/post-buckling configurations.
Ω
0
0
0.05
0.1
0.15
0.2
−w
ðA55 ; A11 ; B11 ; D11 Þ ¼
Z
pre and post-buckling states, the non-linear behavior of a C C beam (L/h = 40, k = 10) is examined in Fig. (12). Apparently, the critical point phenomenon can be postponed by applying the external voltage. For the constituents of this study, applying positive voltage to the electrodes, generates the extra axial compressive electrical force and critical temperature occurs sooner, while applying negative voltage affects vice versa. 5. Conclusion To assess the effects of direct/inverse piezoelectric effects, composition rule, and the boundary conditions on the response of a monomorph FGPM beam with rectangular cross section under the in-plane thermal/electrical loads, a Ritz-based finite elements formulation is developed. First order shear deformation theory of Timoshenko combined with the von-Karman type of geometrical
ðQ 55 ; Q 11 ; ZQ 11 ; Z 2 Q 11 ÞdZ
h=2
mid
Fig. 11. The effect of the applied actuator voltage on the dimensionless fundamental frequency versus the mid-point dimensionless deflection of S S FGPM beam (L/h = 40, k = 0). The curves associated to V0 = +500 V and V0 = 500 V are overlapped.
h=2
e A31 ; Be31 ¼
Z
h=2
e D15 ; De31 ; Ee31 ¼ e e F 11 ; F 33 ; He33 ¼
Appendix B
c12 ¼
h L
c13 ¼
B11 A11 h
c14 ¼
Ae31 U0 L A11 h
ðe31 b sinðbZÞ; Ze31 b sinðbZÞÞdZ
h=2
Z
h=2
ðe15 cosðbZÞ; e31 ; Ze31 ÞdZ h=2
Z
h=2
h=2
2 k11 cos2 ðbZÞ; k33 b2 sin ðbZÞ; k33 b sinðbZÞ dZ
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c22 ¼
K s A55 A11
K 12 ij ¼
K s A55 L A11 h
K 13 ij ¼
K s De15 U0 A11 h
K 14 ij ¼
B11 h D11
K 21 ij ¼
c32 ¼
K s A55 Lh D11
K 22 ij ¼
c33 ¼
K s A55 L2 D11
K 23 ij ¼
c23 ¼ c24 ¼ c31 ¼
c34 ¼ c34 ¼ c41 ¼ c42 ¼
Be31
K s De15
LU0
K 24 ij
D11 Be31 L
U0
Ae31 h F e33 LU0
K 32 ij ¼
De15 h K 33 ij ¼
c43 ¼ c44 ¼
F e11 F e33 L2
c43 ¼
De15 e F 33 L 0
K 34 ij ¼
U
K 41 ij ¼
U
K 42 ij ¼ 2
ðg11 ; g13 ; g22 ; g31 ; g33 Þ ¼ L
Appendix C
M 11 ij M 13 ij
¼
le
¼
¼
Z le
Z le
M 33 ij ¼ K 11 ij
Z le
M 22 ij ¼ M 31 ij
Z
¼
Z le
I0 I1 I0 I1 h I2 ; ; ; ; A11 A11 h A11 D11 D11
K 43 ij ¼ K 44 ij ¼
g11 W
1 i
1 j
W dx F 1i ¼
g13 W
1 i
3 j
W dx F 2i ¼
g22 W2i
W2j dx F 3i ¼
g31 W
3 i
g33 W3i
Z
W
le
1 i
W3j dx ;x
le
Z le
Z
dx
W3j
;x
dx
;x
c14 W1i W4j dx ;x
le
c12 w;x W2i
W1j
;x
dx
;x
Z 1 2 2 2 2 dx c12 w;x Wi Wj þ ðkVT c12 þ c22 Þ W2i W2j ;x ;x ;x ;x le 2 Z le
Z
c12 c13 w;x W2i
W3j
;x
;x
þ
c22 2 3 W W dx c12 i ;x j
F 4i ¼
;x
le
c31 W3i
Z le
W1j
;x
1 2
le
W3i
W3j
;x
;x
;x
;x
dx
;x
c31 c12 w;x W3i
Z
;x
W2j
;x
dx þ c32 W3i W2j ;x
þ
c32 3 3 W W dx c12 i j
Z 34 W3i W4j þ ðc34 c 34 ÞW3i W4j dx c ;x
le
Z
le
c41 W4i W1j
Z le
le
Z le
Z le
Z le
Z le
;x
dx
;x
;x
W2j
1 dx þ c12 c41 w;x W4i W2j ;x ;x 2
c43 W4i W3j c43 ;x
Z le
c42 W4i
Z
W dx
1 j
W
;x
1 j
c13 W1i
le
K 31 ij ¼
De15 Be31 F e33 L 0
le
Z
1 c12 w;x W1i W2j dx ;x ;x 2
Z dx ¼ c12 c14 w;x W2i W4j c24 W2i W4j
D11
F e33 L2 U0
Z
c44 W4i
;x
W4j
;x
c42 4 3 dx W W c12 i j ;x
þ W4i W4j dx
kVT W1i dx ;x
kq W2i dx kVT W3i dx ;x
ðkV kT Þ W4i dx
where le is the element length. The inertia elements which are not specified above, are equal to zero.
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M. Komijani et al. / Composite Structures 98 (2013) 143–152
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