Science in China Series G: Physics, Mechanics & Astronomy 008 SCIENCE IN CHINA PRESS Springer-Verlag www.scichina.com phys.scichina.com www.springerlink.com Three-dimensional analytical solution for a transversely isotropic functionally graded pieoelectric circular plate subject to a uniform electric potential difference LI XiangYu, DING HaoJiang & CHEN WeiQiu Department of Civil Engineering, Zhejiang University, Hanghou 31007, China This paper studies the problem of a functionally graded pieoelectric circular plate subjected to a uniform electric potential difference between the upper and lower surfaces. By assuming the generalied displacements in appropriate forms, five differential equations governing the generalied displacement functions are derived from the equilibrium equations. These displacement functions are then obtained in an explicit form, which still involve four undetermined integral constants, through a step-by-step integration which properly incorporates the boundary conditions at the upper and lower surfaces. The boundary conditions at the cylindrical surface are then used to determine the integral constants. Hence, three-dimensional analytical solutions for electrically loaded functionally graded pieoelectric circular plates with free or simply-supported edge are completely determined. These solutions can account for an arbitrary material variation along the thickness, and thus can be readily degenerated into those for a homogenous plate. A numerical example is finally given to show the validity of the analysis, and the effect of material inhomogeneity on the elastic and electric fields is discussed. functionally graded material, pieoelectric material, circular plates, uniform electric potential, direct displacement method The time-line of humankind is located at the dawn of a new age, namely, the Smart Materials Age [1]. Now, the functionally graded pieoelectric materials (FGPMs), as intelligent materials, have been used extensively in applications of sensors and actuators in the micro-electro-mechanical system (MEMS) [,3] and smart structures [4-9], especially in the medical and aerospace industry [10]. Generally, FGPMs are inhomogeneous with material properties varying spatially in a continuous manner, hence exhibiting more attractive features when compared to conventional laminated ma- Received August 4, 006; accepted August 6, 007 doi: 10.1007/s11433-008-0100- Corresponding author (email: dinghj@ju.edu.cn) Supported by the National Natural Science Foundation of China (Grant Nos. 104710 and 1043030) and the Specialied Research Fund for the Doctoral Program of Higher Education of China (Grant No. 0060335107) Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115
terials. Consequently, more and more attention from researchers in various relevant fields has been paid to the response of the FGPM structures to externally applied loads. The transfer matrix method incorporating the asymptotic expansion technique has been extensively adopted to analye the response of the FGPM structures by Cheng et al. [11-15]. Introducing two displacement functions as well as two stress functions, Chen and Ding [16,17] established a new state-space model (SSM) to examine the bending and free vibration of transversely isotropic FGPM plates. Chen et al. [18] obtained a three-dimensional analytical solution of a functionally graded pieoelectric spherical shell rotating at a constant angular velocity. Wu et al. [19] derived a high-order theory for functionally graded pieoelectric shells based on the generalied Hamilton s principle. Zhong and Shang [0], for the first time, presented a 3D exact analysis of simply supported FGPM rectangular plates via SSM when the material constants vary exponentially along the thickness of the plate. Chen et al. [1] employed SSM along with a layerwise approximation model to analye the free vibration of a functionally graded pieoelectric hollow cylinder filled with compressible fluid. Pan and Han [] obtained the Green s functions for transversely isotropic pieoelectric functionally graded multilayered half space by virtue of two systems of vector functions and the propagator matrix method. Recently, the Stroh-like formulae have been developed to study the mechanical and electric response of simply supported plates [3,4]. However, to the authors knowledge, no analytical solution pertinent to FGPM circular plates has been reported yet. This paper develops a 3D analysis of electrically loaded transversely isotropic FGPM circular plates with free or simply supported edge by virtue of the direct displacement method, which expresses the displacements and electric potential in terms of appropriate polynomials of r, the radial coordinate, with coefficients being undetermined functions of, the axial coordinate. The derived analytical solutions satisfy exactly the boundary conditions on the upper and lower surfaces of the plate, while approximately the circumferential boundary conditions in the Saint Venant s sense. Provided certain integrable conditions and the positive definiteness of energy function are ensured, the material properties of FGPM plates can be of arbitrary functions of. Hence the derived solutions apply to homogeneous plates naturally, for which the elastic and electric fields can be expressed in explicit forms. A numerical example of a particular FGPM circular plate, with only two material constants being functions of the thickness coordinate, is finally considered to clearly show the effect of material inhomogeneity on the elastic and electric fields in the plate. 1 Basic equations and the direct displacement method When referred to the cylindrical coordinates ( r, θ, ), the basic equations for the axisymmetric problem of a transversely isotropic FGPM body are 1 1 σ + τ + r σ σ = 0, τ + r τ + σ = 0, (1) ( ) rr, r, r θ rr, r, 1 Drr, r Dr D, 0, 1 1 r = c11u, r + c1r u+ c13 w, + e31 φ, = c1 u, r + c11r u+ c13 w, + e31 φ, 1 = c13 u, r + r u + c33 w, + e33φ,, τr = c44( u, + w, r) + e15 φ, r, ( ) + + = () σ, σ θ, σ 1 ( r ) D = e ( u + w ) ε φ, D = e u + r u + e w ε φ, (4) r 15,, r 11, r 31, 33, 33, where σ r, σθ, σ and τ r are the stress components; D r and (3) D are the electric displacement LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115 1117
components; u and w are the displacement components in r-direction and -direction, respectively; φ is the electric potential; the comma denotes differentiation with respect to the indicated variable. c ij, e ij and ε ij are the elastic, pieoelectric and dielectric coefficients, respectively. In this paper, we consider functionally graded materials, whose material coefficients in eqs. (3) and (4) are functions of, i.e., c = c ( ), ε = ε ( ) and e = e ( ). For homogeneous materials, these ij ij ij ij coefficients are constant. We start the analysis directly from the generalied displacements (mechanical displacements and electric potential) by assuming that u = ru (5) ij ( ) 1, ( ) ( ) φ φ ( ) 0 0 ij w= w + r w, =, (6) i where u 1 ( ) and w ( ) are referred to as displacement functions, and φ 0 ( ) as electric functions. Substituting eqs. (5) and (6) into eqs. (3) and (4) yields σ r = ( c11 + c1 ) u1 + c13w0, + e31φ 0, + r c13w,, σθ = ( c11 + c1 ) u1 + c13w0, + e31φ 0, + r c13w,, σ = c u + c w + e φ + r c w, τ = rc u + w, 1 33 0, 33 0 r ( ) ( ) ε φ 13, 33, 44 1, D = re u + w, D = e u + e w + r e w. (8) r 15 1, 31 1 33 0, 33 0, 33, Introducing eqs. (7) and (8) into eqs. (1) and (), respectively, we obtain the following equations governing the generalied displacement functions: e w = 0, (9) ( 33, ), ( 33, ) 0,, c44 ( u1, w ) + + c, 13w, = 0, ( e31u1 e33w0, ε33φ0, ) e15 ( u1, ), w ( c13u1 c33w0, e33φ 0, ) c44 ( u1, w ) (7) c w = (10) (11) + + + = 0, (1) + + + + = 0. (13) Circular plate subject to uniform electric potentials, Consider a circular plate with radius a and height h, subject to uniform electric potential k 1 on the upper surface = h and k on the lower surface = h, as shown in Figure 1. The boundary conditions on the surfaces =± h are = h : σ = 0, τr = 0, φ = k1, (14) and = h : σ = 0, τ = 0, φ = k. (15) Substitution of the expressions for in eqs. (14) and (15) leads to σ, r τ r and φ in eqs. (7) and (6) into the boundary conditions 1118 LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115
and =± h ; Figure 1 FGPM circular plate subject to uniform electric potentials at = ±h/. ( ) φ ( ) φ h = k, h = k, (16) 0 1 0 w, = 0, (17) c u + c w + e φ = 0, (18) 13 1 33 0, 33 0, u + w = 0. (19) 1, Integrating eq. (10) once from the lower limit /, using eq. (17) and noticing c 33 > 0, we obtain w, = 0, (0) which satisfies eq. (9) automatically. w is thus obtained from eq. (0): w = a (1) 1, where a 1 is an integral constant. Introducing eq. (0) into eq. (11), integrating once from the lower limits /, using the boundary condition eq. (19) and noticing 44 0 c >, we obtain u + w = 0. () 1, Substituting eq. (1) into eq. () and integrating once from the lower limit / give u = a + a, (3) 1 1 where a is an integral constant. Introducing eq. () into eqs. (1) and (13), integrating them and utiliing the boundary condition (18) yield c u + c w + e φ = 0, (4) 13 1 33 0, 33 0, e u + e w ε φ = a, (5) 31 1 33 0, 33 0, 3 where a 3 is an integral constant. Solving eqs. (4) and (5) simultaneously and making use of eq. (3) give rise to w = 4 a F( ) a F ( ) + a F ( ), (6) where 0, 1 1 3 3 φ = 4 ag( ) a G ( ), (7) 0, 1 1 1 33 33ε33 3 33 J = e + c, F ( ) = J e, 1 1 1 = 31 33 + 13ε33 = 31 33 + 13ε33 F( ) J ( e e c ), F ( ) J ( e e c ), 1 1 1 = 13 33 31 33 = 13 33 31 33 G ( ) J ( c e e c ), G ( ) J ( c e e c ). (8) LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115 1119
Now w 0 and φ 0 can be obtained without any difficulty by integrating eqs. (6) and (7) from the lower limit /: where a 4 and a 5 are integral constants, and w = 4 a f ( ) a f ( ) + a f ( ) + a, (9) 0 1 1 3 3 4 φ = 4 ag( ) ag ( ) + a, (30) 0 1 1 5 gi( ) = Gi( ξ) d ξ, ( i = 1,), (31) f j( ) = Fj( ξ) d ξ, ( j = 1,,3). (3) Introducing eq. (30) into the boundary condition (16), we obtain a 5 1, (33) 4 ag 1 1( h) ag ( h) = k k1 = k, (34) where k is the difference between the electric potentials on the upper and lower surfaces. It is seen that there are totally 5 integral constants ai ( i = 1,,,5), among which a 5 has been fixed and a 1 and a should satisfy eq. (34); boundary conditions at r = a should be employed to fix the four constants ai ( i = 1,,3,4) in addition to eq. (34). It should be noted that a 4 corresponds to a rigid-body translation in the axial direction and a 5 is related to a reference electric potential. Both a 4 and a 5 exert no influence on the distribution of stresses and electric displacement. Substituting the expressions for displacement functions and electric functions into eqs. (5) (8) yields u = r a a, (35) ( ) 1 w= a 1 r 4 f1( ) + af( ) + a3f3( ) + a4, (36) φ = 4 ag 1 1( ) ag ( ) + k1, (37) σ = τ r = 0, σr = σθ = a1[ c13f1( ) + e31g1( ) ( c11 + c1) ] (38) a c F ( ) + e G ( ) ( c + c ) + a c F ( ), [ ] 13 31 11 1 3 13 3 Dr = 0, D = a3. (39) From eqs. (38) and (39), we can observe that: (1) σ r and σ θ are independent of r, being a function of variable only; () The axial electric displacement is constant while the radial electric displacement vanishes in the present problem. It is easy to obtain the expressions for the radial resultant force N and the bending moment M by virtue of σ r in eq. (38): where h σ rd 1 01 0 3 03, (40) N = a N a N + a N h σ rd 1 11 1 3 13, (41) M = a N a N + a N [ ] h i i1 = 13 1 + 31 1 11 + 1 N c F ( ) e G ( ) ( c c ) d, 110 LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115
[ ] h i i = 13 + 31 11 + 1 N c F ( ) e G ( ) ( c c ) d, h i i3 13 3 N = c F ( )d, ( i = 0,1). (4) 3 The determination of remaining integral constants First, consider an FGPM circular plate with free edge. The boundary conditions at r = a are Na ( ) = 0, Ma ( ) = 0, τ r = 0, Dr = 0, (43) where the last two conditions have been satisfied automatically as seen from eqs. (38) and (39). Introducing eqs. (40) and (41) into the first two conditions in eq. (43), we arrive at an 1 01 an 0 + an 3 03 = 0, (44) an 1 11 an 1 + an 3 13 = 0. (45) Solving eqs. (34), (44) and (45) simultaneously, we obtain ai = kai, ( i = 1,,3), (46) where A1 = ( N03N1 N13N0) (4 D), A = ( N03N11 N13N01) ( D), A3 = ( N0N11 N01N1 ) ( D), (47) D= g1( h ) ( N03N1 N13N0 ) g( h ) ( N03N11 N13N01). Substituting eq. (46) into eqs. (35)-(39), we can derive the expressions for displacements, electric potential and electric displacements as follows: u = rk A A, (48) ( ) 1 { } w= k A1 r + 4 f1( ) Af( ) + A3f3( ) + a4, (49) φ = k[ Ag 1 1( ) Ag( ) ] + k1. (50) σ = τ r = 0, σr = σθ = k{ A1[ c13f1( ) + e31g1( ) ( c11 + c1) ] (51) A c F ( ) + e G ( ) ( c + c ) + A c F ( ). [ ] } 13 31 11 1 3 13 3 Dr = 0, D = ka3. (5) For a simply-supported circular plate, since the axial displacement is constrained at r = a, we may simply take a4 = ka 1 a 4 f1(0) + + kaf(0) ka3f3(0) in eq. (49) while eqs. (48) and (50)-(5) keep unaltered.. For homogeneous materials, i.e., cij = const, eij = const and ε ij = const, all functions involved become polynomials of. Calculation according to eq. (47) gives rise to the explicit expressions for Ai ( i = 1,, 3). The corresponding expressions for stresses, displacements, electric potential and electric displacements are u = rk ( Ch), w= k( + h ) C5 ( Ch) + a4, (53) φ = k h+ 1 + k, (54) ( ) 1 LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115 111
where 1 σ = τr = σr = σθ = 0, (55) D = 0, D = C k ( c C C h), (56) r 4 13 3 1 1 1 31 33 + 13ε33 = 13 33 31 33 3 = 33 C = J ( e e c ), C J ( c e e c ), C J e, [ e ] 1 13C1 31 11 + 1 5 = 13 11 + 1 31 C4 = c + e C ( c c ), C c ( c c ) C. Comparison of eqs. (48)-(5) with eqs. (53)-(56) reveals that: (1) The radial displacement of a homogeneous plate, varying linearly with r, is independent of, while for an inhomogeneous plate, the radial displacement depends upon in a linear manner; () The deflection of a homogeneous plate w, being a linear function of, does not vary with r, whereas in the FGPM case, the expression of deflection involves r. Hence the neutral surface ( = 0) of the FGPM plate becomes a curved surface after deformation; while that of a homogeneous plate remains a plane parallel to the upper and lower surfaces; (3) All stresses in a homogeneous plate vanish, while σ r and σ θ generally are not ero in an FGPM plate. Similarly, if we assign a4 = kc5 (4 C), eqs. (53)-(56) apply to a simply supported plate. 4 Numerical results and discussion We consider a particular form of FGPM as follows: 0 0 0 0 λη+ ( 1) 0 c = c, c = c, c = c, c = c e, c = c, (58) where c 0 ij, 0 e ij and 11 11 1 1 13 13 33 33 44 44 0 ij 0 0 0 31 31 33 33 15 15 (57) e = e, e = e, e = e, (59) 0 0 ( 1) ε ε, ε ε e λη + = =, (60) 11 11 33 33 ε are the material coefficients at the upper surface = h, 1 η = h 1 is the dimensionless thickness coordinate, and λ is the gradient index. Particularly when λ = 0, the material becomes homogenous. For this particular FGPM, all elastic and electric field quantities can be obtained explicitly, however, they are omitted here for the sake of simplicity. The numerical calculation is carried out for a simply supported circular plate, and the following dimensionless quantities are introduced: r σ r D w φ ξ =, 0 ξ 1, σξ =, D, W,. 0 η = = Φ = (61) a c 0 0 c ε h k 11 11 33 We take the electric potentials k 1 = 0 V, k = 100 V, the radius of the plate a = 1 m, and the height h = 0.1 m. The material property at = h is identical to PZT-5H [5,6], with material constants shown in Table 1. Table 1 Material properties of PZT-5H Property Elastic (10 10 N/m ) Pieoelectric (C/m ) Dielectric (10 11 F/m) PZT-5H 0 0 0 0 0 11 1 13 33 44 c = 1.6, c = 5.5, c = 5.3, c = 11.7, c = 3.53 0 0 0 15 31 33 e = 17.0, e = 6.5, e = 3.3 0 0 11 ε33 ε = 1510, = 1300 11 LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115
As known from the earlier analysis, σ ξ is independent of r, being a function of only, and its distribution along the thickness is depicted in Figure, which shows that the curve for λ = 1 tends to be parabolic, the one for λ = 0 is a horiontal straight line, while that for λ = 1 decreases with η. The axial electric displacement ( D η ) is a constant as shown in eq. (39), but varies with parameters, as shown in Figure 3. It is seen that D η for λ = 0 is larger than those for λ =± 1. Figure Dimensionless radial stresses of a circular plate subject to uniform electric potentials at η = ±1/. Figure 3 Dimensionless normal electric displacements of a circular plate subject to uniform electric potentials at η = ±1/. Figure 4 shows the distribution curves of the dimensionless deflection W on the middle plane. From Figure 4, we can see that the curve for λ = 0 is a straight line, i.e., W ( ξ ) = 0 ; the one for λ = 1 is concave and W ( ξ ) 0 ; while the one for λ = 1 is convex and W ( ξ ) 0. The distribution of the dimensionless electric potential Φ at the center ξ = 0 along the thickness η is shown in Figure 5, which illustrates that Φ changes in a linear way when λ = 0 and is larger than those corresponding to λ =± 1. Figure 4 Dimensionless normal deflections of a circular plate subject to uniform electric potentials at η = ±1/. Figure 5 Dimensionless electric potentials of a circular plate subject to uniform electric potentials at η = ±1/. LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115 113
5 Conclusions Three-dimensional analytical solutions were derived for the axisymmetric problem of transversely isotropic circular plates made of functionally graded pieoelectric materials. The material constants can vary along the thickness in an arbitrary way and the plate is simply supported or with free edges. The present solutions are independent of the material eigenvalues and can be degenerated into those for homogenous materials. Numerical results show that the material inhomogeneity exerts an important effect on the elastic and electric fields in the plate. Hence, optimum design may be performed to obtain the desired results through the proper choice of material property variation along the thickness. Since no ad hoc hypotheses on the distributions of elastic and electric fields are introduced, the present analytical solutions could provide a useful meaning of checking the validity of various approximate theories and numerical methods. 1 Gandhi M V, Thompson B S. Smart Materials and Structures. London: Chapman & Hall, 199 Yee Y, Nam H J, Lee S H, et al. PZT Actuator micromirror for finite-tracking mechanism of high-density optical data storage. Sens Actuat A, 001, 89: 166-173[DOI] 3 Wang S, Li J F, Wakabayashi K, et al. Lost silicon mold process for PZT microstructure. Adv Mater, 1999, 10: 874-876 4 Taya M, Almajid A A, Dunn M, et al. Design of bimorphpieo-composite actuators with functionally graded microstructure. Sens Actuat A, 003, 107: 48-60[DOI] 5 Wu C C M, Kahn M, Moy W. Pieoelectric ceramics with functionally gradients, a new application in material design. J Am Ceram Soc, 1999, 79: 809-81 6 Ichinose N, Miyamoto M, Takahashi S. Ultrasonic transducers with functionally graded pieoelastic ceramics. J Euro Ceram Soc, 004, 4: 1681-1685[DOI] 7 Zhu X H, Meng Z Y. Operational principle, fabrication and displacement characteristic of a functionally gradient pieoelectric ceramic actuator. Sens Actuat A, 1995, 48: 169-176[DOI] 8 Yan W, Chen W Q. Electro-mechanical response of functionally graded beams with imperfectly integrated surface pieoelectric layers. Sci China Ser G-Phys Mech Astron, 006, 49 (5): 513-55[DOI] 9 Yu T, Zhong Z. Bending analysis of functionally graded pieoelectric cantilever beam. Sci China Ser G-Phys Mech Astron, 007, 50(1): 97-108 10 Bernhard A P F, Chopra I. Trailing edge flag activated by a pieo-induced bending-torsion coupled beam. J Am Helicopter Soc, 1999, 44: 3-15 11 Cheng Z Q, Lim C W, Kitipornchai S. Three-dimensional exact solution for inhomogeneous and laminated pieoelectric plates. Int J Solids Struct, 1999, 37: 145-1439 1 Cheng Z Q, Batra R C. Three-dimensional asymptotic scheme for pieothermoelastic laminates. J Thermal Stresses, 000, 3: 95-110[DOI] 13 Cheng Z Q, Lim C W, Kitipornchai S. Three-dimensional asymptotic approach to inhomogeneous and laminated pieoelectric plates. Int J Solids Struct, 000, 37: 3153-3175[DOI] 14 Cheng Z Q, Batra R C. Three-dimensional asymptotic analysis of multi-electroded pieoelectric laminates. AIAA J, 000, 38(): 317-34 15 Reddy J N, Cheng Z Q. Three-dimensional solutions of smart functionally graded plates. J Appl Mech, 001, 68: 34-41 [DOI] 16 Chen W Q, Ding H J. Bending of functionally graded pieoelectric rectangular plates. Acta Mech Solid Sin, 000, 13(4): 114 LI XiangYu et al. Sci China Ser G-Phys Mech Astron Aug. 008 vol. 51 no. 8 1116-115
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