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Exact static solutions to piezoelectric smart beams including peel stresses
International Journal of Solids and Structures 39 (2002) 4677–4695 www.elsevier.com/locate/ijsolstr
Exact static solutions to piezoelectric smart beams including peel stresses I: Theoretical formulation Quantian Luo, Liyong Tong
*
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia Received 14 November 2001; received in revised form 20 May 2002
Abstract This paper presents exact static solutions to smart beams with perfectly bonded or partially debonded piezoelectric (PZT) actuators and sensors including peel stresses. When a PZT patch is bonded on the surface of a beam, both shear and peel stresses exist in the adhesive between the PZT patch and the host beam. In Part I, non-dimensional differential governing equations for infinitesimal elements of the PZT patch and the host beam are formulated firstly, and then the ordinary differential equation (ODE) of the coupled shear and peel stresses are completely solved analytically. The exact solutions are applicable to smart beams with PZT actuators and sensors. When the PZT patches are used as actuators, the solutions give the formulations for the energy transferring shear and peel stresses in the adhesive actuated by the applied voltages, and for the actuated stress resultants and displacements in the host beam. When the PZT patches are used as sensors, the solutions give the shear and peel stresses in the adhesive caused by the loadings, and also the sensing electric charges. When PZT patches are not perfectly bonded to the host beam, i.e., partially debonded, the developed solutions can be tailored to the case of edge debondings, i.e., PZTs with the shortened length. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectric actuator and sensor; Peel stress; Coupled differential equation
1. Introduction A smart structure with piezoelectric (PZT) actuators or sensors has drawn a considerable interest in structural performance control due to the coupled mechanical and electrical properties of PZT materials. PZTs are normally bonded to or embedded in the host structure in structural applications such as vibration and shape control. Crawley and de Luis (1987) developed a theoretical framework for modeling extensional and bending deformations of a beam with PZT actuators, which has been frequently cited and widely used
*
Corresponding author. Tel.: +61-2-9351-6949; fax: +61-2-9351-4841. E-mail address: [email protected] (L. Tong).
0020-7683/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 0 2 ) 0 0 3 8 3 - 9
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as the theoretical fundamentals. Their theory assumed that PZT patches are perfectly bonded to the host structures and only shear stresses exist in the adhesive layer. Using the similar assumptions, Crawley and Lazarus (1991) applied the theory to plates with perfectly bonded PZTs. Im and Atluri (1989) formulated governing equations for a beam column linked with piezo-actuators subjected to general loading, in which the peel stresses were considered. However, they believed that the peel stresses could be ignored for most cases and the peel stresses were neglected in their subsequent analysis. Shi and Atluri (1990) utilized these results for active control of space frame vibrations. They also assumed that PZT patches were perfectly bonded to the host structures. Robbins and Reddy (1991) modeled the adhesive layer as a beam using a layer-wise displacement theory. Although a beam model for the adhesive layer could be used to study the peel effects, Goland and Reissner (1944) showed that the thin flexible joint layer would not bear bending moments and behave like a spring with the shear and peel stiffness. Anderson et al. (1977) described the analysis and tests for this adhesive behavior. Post et al. (1988) and Dillard et al. (1988) testified this characteristic of the adhesive layer by photo-elastic and mechanical experiments respectively. Tong and Steven (1999) gave systematic description for mechanical behaviors of structural loaded joints. All of these works modeled the adhesive layer as the spring system with the shear and peel stiffness. Suhir (1986, 2000) also employed this adhesive model for analyzing thermal stresses in the adhesive. The classic theory of Goland and Reissner (1944) has been shown to predict the reasonable stress distributions in the thin adhesive layer. Kim and Jones (1996) conducted an analytical and experimental investigation to identify the effects of delamination on the smart beam with PZT. Seeley and Chattopadhyay (1998) also performed the experimental investigation of composite beams with PZT actuation and debonding, and then they (Seeley and Chattopadhyay, 1999) further studied debonding effects of the smart beams using a refined higher order finite element analysis. Wang and Meguid (2000) utilized the solutions of a whole plane and a half plane subjected to a concentrated horizontal force for analyzing debonding effects of the embedded and bonded PZT patch. These models did not model the adhesive layer and thus may not be used to investigate the peel effects. Tong et al. (2001) and Sun et al. (2001) modeled the adhesive layer as a shear and peel spring system and studied the effects of debondings, stresses in the adhesive layer and stress resultants for smart beams. In their analysis, the multi-segment shooting method was used to solve the governing equations. In the present study, based on the classic theory in bonding joints (Tong and Steven, 1999), a theoretical model for a PZT composite beam including adhesive layers is developed. In this model, PZT patches and the host beam are modeled as Euler–Bernoulli beams, and the adhesive layer is modeled as a continuous spring with the shear and peel stiffness. Using this model, differential equations of the equilibrium are formulated. Incorporating the constitutive relations of PZT, the adhesive layer, and the host beam with the equilibrium equations, we can obtain the differential equations of the coupled shear and peel stresses. The governing equations can be completely solved analytically, and the shear and peel stresses in the adhesive are exactly obtained. Based on analytical solutions of the shear and peel stresses, we can derive the actuated internal forces and displacements of the host beam as well as the sensing electric charge of the PZTs. In this analytical model, the edge debondings are also studied by simply considering a shortened PZT length.
2. Basic differential equation Consider a PZT patch bonded to the host beam on the top surface, as shown in Fig. 1, the free body diagrams for the infinitesimal elements of the PZT, adhesive layer and host beam are depicted in Fig. 2. In the light of static equilibrium conditions, we can perform the following formulation, in which the unit width of the whole beam with the rectangular cross section.
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Fig. 1. PZT composite beam.
Fig. 2. Stresses and forces on the infinitesimal elements of the substructures: (a) the PZT infinitesimal element, (b) the adhesive layer element and (c) the host beam element.
The following parameters are introduced for the non-dimensional formulations: 8 Ni Qi Mi Ei s r ui > > ; Qni ¼ ; Mni ¼ ; Eni ¼ ; sn ¼ ; rn ¼ ; uni ¼ ; < Nni ¼ 2 Ga Ga Ga L Ga L Ga L Ga L wi ti x L Ea t1 h > > : wni ¼ ri ¼ ; n ¼ ; ra ¼ ; rav ¼ ra ; Rta ¼ ; Rht ¼ ði ¼ 1; hÞ L ta t1 L L Ga ta
ð1Þ
The subscripts 1, h, and a in the above and all subsequent equations refer to the PZT patch, the host beam and the adhesive layer. In all above equations, Ei , and ti (ti ¼ t1 , h) are the elastic moduli and the thickness of the sub-structure; ui and wi (i ¼ 1; h) are the extensional and flexural displacements; e31 and V1 are a PZT coupling constant of the PZT and the applied voltage; Ea and Ga are the Young’s and shear moduli of the adhesive layer. The remaining mathematical symbols are defined in Fig. 2.
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As shown in Fig. 2, the non-dimensional ordinary differential equations (ODEs) of the equilibrium are: dNn1 þ sn ¼ 0; dn
dQn1 þ rn ¼ 0; dn
dMn1 1 þ r1 sn Qn1 ¼ 0 2 dn
ð2Þ
dNnh sn ¼ 0; dn
dQnh rn ¼ 0; dn
dMnh 1 þ rh sn Qnh ¼ 0 2 dn
ð3Þ
By employing the mechanical–electrical relations of PZT (Holland and EerNisse, 1969; Chee et al., 1998) and the Euler–Bernoulli beam theory (Timoshenko and Gere, 1972), we can obtain the following constitutive relations for the PZT and the host beam: Nn1 ¼ En1 r1
dun1 e31 V1 þ ; dn Ga L
Nnh ¼ Enh rh
dunh ; dn
Mn1 ¼
Mnh ¼
1 d2 wn1 En1 r13 12 dn2
ð4Þ
1 d2 wnh Enh rh3 12 dn2
ð5Þ
Referring to Fig. 2 and the definition given by Goland and Reissner (1944), we have the constitutive relations for the adhesive layer: Rta dwnh dwn1 Rht sn ¼ ra ðunh un1 Þ þ þ ð6Þ 2 dn dn rn ¼ rav ðwnh wn1 Þ
ð7Þ
Differentiating Eqs. (6) and (7), into which Eqs. (2)–(5) are substituted, we can obtain the following coupled differential equation: d 3 sn dsn þ kn2 rn ; ¼ kn1 3 dn dn
d4 rn dsn kn4 rn ¼ kn3 4 dn dn
ð8Þ
where, kn1 ¼
4Rta r12
rav kn3 ¼ kn2 ; ra
1 1 þ ; En1 Enh Rht 12rav Rta kn4 ¼ ra r14
kn2 ¼
6Rta r13
1 1 En1 Enh R2ht
1 1 þ En1 Enh R3ht
9 > > > = > > > ;
ð9Þ
When Enh R2ht En1 ¼ 0, kn2 and kn3 in Eq. (8) become zero, and thus governing equation (8) for the shear and peel stresses becomes decoupled and can be simplified as d 3 sn dsn ; ¼ b2uc1 3 dn dn
d4 rn ¼ b4uc2 rn dn4
ð10Þ
in which, b2uc1 ¼ kn1 ;
b4uc2 ¼ kn4
ð11Þ
The decoupling condition Enh R2ht En1 ¼ 0 can be rewritten as Eh h2 =E1 t12 ¼ 1. Evidently, when both the PZT and host beam have the same Young’s moduli and thickness, the decoupling condition is satisfied. This case was referred to as the balanced single-lap joints by Goland and Reissner (1944) and Tong and Steven (1999), for which the decoupled equation (8) was solved for the selected loadings and boundary conditions.
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3. Solution to the coupled equation (8) To solve the coupled governing equation (8), let’s assume sn ¼ A ebn and rn ¼ B ebn and then substitute into Eq. (8), we have ðb3 kn1 bÞA kn2 B ¼ 0;
kn3 bA þ ðb4 þ kn4 ÞB ¼ 0
ð12Þ
The condition of non-trivial solutions to the shear and peel stresses requires ½b6 kn1 b4 þ kn4 b2 ðkn1 kn4 kn2 kn3 Þb ¼ 0
ð13Þ
which becomes k3 kn1 k2 þ kn4 k ðkn1 kn4 kn2 kn3 Þ ¼ 0;
where; k ¼ b2 ; or b ¼ 0
ð14Þ
Roots of the above polynomial can be analytically solved (Uspensky, 1948), and there always exist one real number root and two conjugate complex number roots for the most combinations of typical material properties and geometric configurations. Therefore, we can express solutions to the shear and peel stresses as sn ¼ A1 sinh b1 n þ A2 cosh b1 n þ ðA3 sinh b2 n þ A4 cosh b2 nÞ sin b3 n þ ðA5 sinh b2 n þ A6 cosh b2 nÞ cos b3 n þ A7 rn ¼ B1 sinh b1 n þ B2 cosh b1 n þ ðB3 sinh b2 n þ B4 cosh b2 nÞ sin b3 n þ ðB5 sinh b2 n þ B6 cosh b2 nÞ cos b3 n þ B7
ð15Þ
ð16Þ
In which Ai and Bi (i ¼ 1–7) are the integration constants to be determined using the boundary conditions; b1 , b2 , and b3 are determined by solving Eq. (13). Combining both equations in Eq. (8), by eliminating the shear stress, we have an ODE of six-order for the peel stress. Substituting Eq. (16) into the six-order ODE and noting Eq. (13), we find B7 ¼ 0. The first equilibrium equation shown in Eq. (2) defines a supplementary equation for the shear stress: Z 1 sn ðnÞ dn ¼ Nn1II Nn1I ð17Þ 1
where, Nn1II and Nn1I are the non-dimensionalized axial forces applied at the PZT edges (see Fig. 3). Fig. 3 illustrates that one PZT patch is bonded on the top surface of the host beam. In Fig. 3, N1k , M1k and Q1k (k ¼ I; II) are the applied forces at the edges of the PZT or the internal forces at the debonding edges; Nhk ; Mhk and Qhk (k ¼ I; II) are the internal forces at cross-sections I–I and II–II of the host beam, which can be solved by the equilibrium conditions for the statically-determined structures.
Fig. 3. Boundary conditions of the host beam with the partially bonded PZT.
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Referring to Fig. 3, we directly obtain the non-dimensional forces in the boundaries using Eq. (1), and then, substituting the non-dimensional forces into the constitutive and equilibrium equations of the PZT patch and the host beam, we can solve the displacements at n ¼ 1: 9 dun1k Nn1k d2 wn1k 12Mn1k d3 wn1k 6snk 12Qn1k > > ¼ þ ee ; ¼ ; ¼ > dn En1 r1 En1 r12 En1 r13 En1 r13 = dn2 dn3 ðk ¼ I; IIÞ ð18Þ > dunhk Nnhk d2 wnhk 12Mnhk d3 wnhk 6snk 12Qnhk > > ; ¼ ; ¼ ; ¼ dn Enh rh2 En2 r22 En2 rh3 En2 rh3 dn2 dn3 In which, ee ð¼ e31 V1 =E1 t1 Þ is the electrically actuated strain. Substituting Eq. (18) into the constitutive relations of the adhesive, we obtain the following boundary conditions for solving the governing equation (8): 9 dsn > > ¼ Hnk > > dn > > 2 > = d rn ¼ H mk ðk ¼ I; IIÞ ð19Þ n ¼ 1 : dn2 > > 3 > > d rn > þ kn3 sn ¼ Hqk > > ; dn3 where,
9 Nn1k Nnhk Mn1k Mnhk > > Hnk ¼ ra þ ee þ Rht þ 6Rta > > En1 r1 Enh rh En1 r13 Enh rh3 > > > = Mnhk Mn1k Hmk ¼ 12rav 3 3 > Enh rh En1 r1 > > > > > Qnhk Qn1k > ; Hqk ¼ 12rav 3 3 Enh rh En1 r1
ðk ¼ I; IIÞ
ð20Þ
Solutions to Eqs. (15)–(17) with the boundary conditions defined in Eq. (19) cannot be directly solved, as 12 integration constants cannot be directly determined by six equations of the boundary conditions. This boundary value problem (BVP) of ODE can be solved using the method of elimination. This technique attempts to solve a given system of coupled differential equations by eliminating the coupled items and then reducing a system to the uncoupled higher order equations (Humi and Miller, 1988). For the ODE with constant coefficients in the present study, we can also obtain Eqs. (15)–(17), and then combine the governing equation (8) and the boundary conditions shown in Eq. (19) to obtain the higher order equations of the boundary conditions, which can be used to determine the integration constants. Alternatively, we may substitute Eqs. (15) and (16) into the first formulation or the second one of Eq. (8). In doing so, we can obtain six relations between Ai and Bi , and thus the integration constants can be uniquely solved. Dokumaci (1987) obtained an exact solution for the coupled bending and torsions of uniform beams having single cross-sectional symmetry using this method, which was the first exact solution in real form according to his literature search. Employing this method, Banerjee (1989) derived the coupled bending and torsion dynamic stiffness matrix for beam elements. Using Mathematica, Tanaka and Bercin (1999) solved the free vibration for uniform beams of non-symmetrical cross-section utilizing this method. The above two methods have been used for solving this BVP and the same results are obtained. Due to the space limit, only the solution procedure using the second one is presented here. Substituting Eqs. (15) and (16) into the first formulation of Eq. (8), we find B2 ¼ K1 A1 ;
B3 ¼ K2 A4 þ K3 A5 ;
B6 ¼ K3 A4 þ K2 A5
ð21Þ
B1 ¼ K 1 A2 ;
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B4 ¼ K2 A3 þ K3 A6 ;
ð22Þ
B5 ¼ K3 A3 þ K2 A6
where, K1 ¼
b1 ðb21 kn1 Þ ; kn2
K2 ¼
b2 ½ðb22 3b23 Þ kn1 ; kn2
K3 ¼
b3 ½ðb23 3b22 Þ þ kn1 kn2
ð23Þ
Substituting Eqs. (15) and (16) into Eq. (19), and considering Eq. (17), we can obtain Eqs. (24) and (25): 9 A1 b1 cosh b1 þ A4 ðb2 sinh b2 sin b3 þ b3 cosh b2 cos b3 Þ > > > HnII þ HnI > > > þA5 ðb2 cosh b2 cos b3 b3 sinh b2 sin b3 Þ ¼ > > 2 > > 2 2 2 > > B2 b1 cosh b1 þ B3 ½ðb2 b3 Þ sinh b2 sin b3 þ 2b2 b3 cosh b2 cos b3 > = H þ H mII mI 2 2 ð24Þ þB6 ½ðb2 b3 Þ cosh b2 cos b3 2b2 b3 sinh b2 sin b3 ¼ > 2 > > 3 2 2 2 2 > B2 b1 sinh b1 þ B3 ½ðb2 ðb2 3b3 Þ cosh b2 sin b3 b3 ðb3 3b2 Þ sinh b2 cos b3 Þ > > > > þB6 ½ðb2 ðb22 3b23 Þ sinh b2 cos b3 þ b3 ðb23 3b22 Þ cosh b2 sin b3 Þ > > > > HqII HqI > ; þkn3 ðA1 sinh b1 þ A4 cosh b2 sin b3 þ A5 sinh b2 cos b3 Þ ¼ 2 9 A2 b1 sinh b1 þ A3 ðb2 cosh b2 sin b3 þ b3 sinh b2 cos b3 Þ > > > HnII HnI > > þA6 ðb2 sinh b2 cos b3 b3 cosh b2 sin b3 Þ ¼ > > 2 > > 2 2 2 > > B1 b1 sinh b1 þ B4 ½ðb2 b3 Þ cosh b2 sin b3 þ 2b2 b3 sinh b2 cos b3 > > > H H > mII mI 2 2 > > þB5 ½ðb2 b3 Þ sinh b2 cos b3 2b2 b3 cosh b2 sin b3 ¼ > > 2 > > 3 2 2 2 2 B1 b1 cosh b1 þ B4 ½ðb2 ðb2 3b3 Þ sinh b2 sin b3 b3 ðb3 3b2 Þ cosh b2 cos b3 Þ = ð25Þ þB5 ½ðb2 ðb22 3b23 Þ cosh b2 cos b3 þ b3 ðb23 3b22 Þ sinh b2 sin b3 Þ > > > HqII þ HqI > > þkn3 ðA2 cosh b1 þ A3 sinh b2 sin b3 þ A6 cosh b2 cos b3 þ A7 Þ ¼ > > > 2 > > > 1 1 > > A2 sinh b1 þ A3 2 ðb2 cosh b2 sin b3 b3 sinh b2 cos b3 Þ > 2 > b1 > b2 þ b 3 > > > 1 Nn1II Nn1I > > ; þA6 2 ðb sinh b cos b þ b cosh b sin b Þ þ A ¼ 7 2 2 3 3 2 3 2 2 b2 þ b 3 Substituting Eq. (21) into (24), we can obtain a group of the linear algebraic equations: 9 A1 b1 cosh b1 þ A4 ðb2 sinh b2 sin b3 þ b3 cosh b2 cos b3 Þ > > > HnII þ HnI > > > þA5 ðb2 cosh b2 cos b3 b3 sinh b2 sin b3 Þ ¼ > > 2 > > > A1 bK1 cosh b1 þ A4 ðbK2 sinh b2 sin b3 þ bK3 cosh b2 cos b3 Þ > > = HmII þ HmI þA5 ðbK2 cosh b2 cos b3 bK3 sinh b2 sin b3 Þ ¼ > 2 > > > A1 bL1 sinh b1 þ A4 ðbL2 cosh b2 sin b3 bL3 sinh b2 cos b3 Þ > > > > HqII HqI > > > > þA5 ðbL2 sinh b2 cos b3 þ bL3 cosh b2 sin b3 Þ ¼ > ; 2
ð26Þ
where, 9 bK1 ¼ K1 b21 ; bK2 ¼ K2 ðb22 b23 Þ þ 2K3 b2 b3 ; bK3 ¼ 2K2 b2 b3 K3 ðb22 b23 Þ = bL1 ¼ K1 b31 þ kn3 ; bL2 ¼ K2 b2 ðb22 3b23 Þ K3 b3 ðb23 3b22 Þ þ kn3 ; bL3 ¼ K3 b2 ðb22 3b23 Þ þ K2 b3 ðb23 3b22 Þ
ð27Þ
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The integration constants A1 , A4 , A5 can be solved from Eq. (26), and then B2 , B3 , B6 are solved from Eq. (21). Similarly, substituting Eq. (22) into (25) and eliminating A7 , we can obtain the other group of the linear algebraic equations: 9 A2 b1 sinh b1 þ A3 ðb2 cosh b2 sin b3 þ b3 sinh b2 cos b3 Þ > > > HnII HnI > > þA6 ðb2 sinh b2 cos b3 b3 cosh b2 sin b3 Þ ¼ > > > 2 > > = A2 bK1 sinh b1 þ A3 ðbK2 cosh b2 sin b3 þ bK3 sinh b2 cos b3 Þ HmII HmI ð28Þ þA6 ðbK2 sinh b2 cos b3 bK3 cosh b2 sin b3 Þ ¼ > > > 2 > > > A2 bL11 cosh b1 þ A3 ðbL21 sinh b2 sin b3 bL3 cosh b2 cos b3 Þ > > HqII þ HqI kn3 ðNn1II Nn1I Þ > > ; þA6 ðbL22 cosh b2 cos b3 þ bL3 sinh b2 sin b3 Þ ¼ 2 2 where, bL11 ¼ K1 b31 þ kn3
kn3 ðb2 cthb2 b3 ctanb3 Þ þ b23 > > > > kn3 > 2 2 2 2 > ¼ K2 b2 ðb2 3b3 Þ K3 b3 ðb3 3b2 Þ þ kn3 2 ðb thb þ b tan b Þ ; 2 2 3 3 2 b2 þ b3
bL21 ¼ K2 b2 ðb22 3b23 Þ K3 b3 ðb23 3b22 Þ þ kn3 bL22
9 > > > > > > =
kn3 thb1 b1
b22
ð29Þ
The integration constants A2 ; A3 and A6 can be solved from Eq. (28), and then B1 ; B4 ; B5 are solved from Eq. (22) and A7 is solved by the last formulation of Eq. (25). Eqs. (26) and (28) are the linear algebraic equations that can be easily solved. The explicit expressions of all integration constants are given in Appendix A. The integration constants of the decoupled equation (10) for the present boundary conditions are also explicitly given in Appendix A.
4. PZT used as an actuator On the basis of the linear theory, the superposition principle can be applied to the exerted voltage and forces. To investigate performance of a PZT actuator, we only study the mechanical behaviors of the smart beam under action of the applied voltage in PZT. In this case, the boundary conditions become Hnk ¼ ra ee ;
Hmk ¼ 0
and
Hqk ¼ 0;
ðk ¼ I; IIÞ
ð30Þ
4.1. Shear and peel stresses If there are no interior debondings in the adhesive layer, we can find from Eqs. (20) and (25) that, integration constants A2 ; A3 ; A6 ; A7 ; B1 ; B4 , and B5 are equal to zero. In this case, only Eq. (26) needs to be solved. The solutions of the shear and peel stresses are sn ¼ A1 sinh b1 n þ A4 cosh b2 n sin b3 n þ A5 sinh b2 n cos b3 n
ð31Þ
rn ¼ B2 cosh b1 n þ B3 sinh b2 n sin b3 n þ B6 cosh b2 n cos b3 n
ð32Þ
In which, integration constants A1 ; A4 ; A5 are determined from Eq. (26), as given in Appendix A; B2 ; B3 ; B6 are obtained from Eq. (21). Having solved the shear and peel stresses, the actuated internal forces, displacements in the host beam can be obtained. Solutions to the decoupled equation (10) can be simplified as
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sn ¼ A1 sinh buc1 n rn ¼ B1 sinh be n sin be n þ B4 cosh be n cos be n;
9 =
pffiffiffi 2 b ; where; be ¼ 2 uc2
4685
ð33Þ
4.2. Internal forces Substituting the shear and peel stresses into differential forms of the equilibrium equations, we can derive the actuated internal forces in the host beam. The results are: Nnh ¼ Nf s ðnÞ Nf s ð1Þ
ð34Þ
Qnh ¼ ½B2 br1 sinh b1 n þ B3 br2 ðb2 cosh b2 n sin b3 n b3 sinh b2 n cos b3 nÞ þ B6 br2 ðb2 sinh b2 n cos b3 n þ b3 cosh b2 n sin b3 nÞ 1 Mnh ¼ rh ½Nf s ð1Þ Nf s ðnÞ þ ½Mf r ðnÞ Mf r ð1Þ 2
ð35Þ ð36Þ
where, 9 > > > > > þA5 br2 ðb2 cosh b2 n cos b3 n þ b3 sinh b2 n sin b3 nÞ > > > > 2 2 2 2 Mf r ðnÞ ¼ B2 br1 cosh b1 n þ B3 br2 ½ðb2 b3 Þ sinh b2 n sin b3 n 2b2 b3 cosh b2 n cos b3 n = > > þB6 b2r2 ½ðb22 b23 Þ cosh b2 n cos b3 n þ 2b2 b3 sinh b2 n sin b3 n > > > > > 1 1 > > br1 ¼ ; br2 ¼ 2 ; 2 b1 b2 þ b 3 Nf s ðnÞ ¼ A1 br1 cosh b1 n þ A4 br2 ðb2 sinh b2 n sin b3 n b3 cosh b2 n cos b3 nÞ
ð37Þ
The average axial force and bending moment are given by 9 dJs ð1Þ dJs ð0Þ > > Naver ¼ 0 Nnh dn ¼ Nf s ð1Þ > = dn dn R1 > 1 dJs ð1Þ dJs ð0Þ dJr ð1Þ dJr ð0Þ > þ Mf r ð1Þ > Maver ¼ 0 Mnh dn ¼ rh Nf s ð1Þ þ ; 2 dn dn dn dn
R1
ð38Þ
where, Js ðnÞ ¼ A1 b3r1 cosh b1 n þ A4 b3r2 ½b2 ðb22 3b23 Þ sinh b2 n sin b3 n þ b3 ðb23 3b22 Þ cosh b2 n cos b3 n þ A5 b3r2 ½b2 ðb22 3b23 Þ cosh b2 n cos b3 n b3 ðb23 3b22 Þ sinh b2 n sin b3 n
ð39Þ
Jr ðnÞ ¼ B2 b4r1 cosh b1 n þ B3 b4r2 ½ðb42 6b22 b23 þ b23 Þ sinh b2 n sin b3 n 4b2 b3 ðb22 b23 Þ cosh b2 n cos b3 n þ B6 b4r2 ½ðb42 6b22 b23 þ b23 Þ cosh b2 n cos b3 n þ 4b2 b3 ðb22 b23 Þ sinh b2 n sin b3 n
ð40Þ
Because the actuated axial force and the bending moment are the same in the large middle area of the joint, they have been defined as the equivalent axial force and bending moment, which are Nmax ¼ Nnh ð0Þ;
Mmax ¼ Mnh ð0Þ
Eq. (41) defines the non-dimensional equivalence of the actuated forces.
ð41Þ
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Fig. 4. A cantilever beam with the partially bonded PZT.
4.3. The actuated displacements of the host beam In light of the theory of material mechanics (Timoshenko and Gere, 1972), the axial displacement is small as compared with the transverse deflection and thus only the transverse deflection of the beam needs to be solved normally. If necessary, the axial displacement can be simply solved by the constitutive equation as the internal axial force has been found. For the cantilever beam model shown in Fig. 4, which was discussed by Wang and Rogers (1991) without taking into account the adhesive layer, we can obtain 9 a > wnh ¼ 0; 6 n 6 1 > > L > > > > = dwr ð1Þ dws ð1Þ wnh ¼ wr ðnÞ þ ws ðnÞ þ þ ðn þ 1Þ ½wr ð1Þ þ ws ð1Þ; 1 6 n 6 1 ð42Þ dn dn > > > > > > dwr ð1Þ dws ð1Þ > ; þ wnh ¼ 2 n; 1 6 n 6 Lb dn dn where, 12 n2 whr ðnÞ ¼ Mf r ð1Þ Jr ðnÞ ; Enh rh3 2 At n ¼ 1, we have dwr ð1Þ dws ð1Þ wnh ð1Þ ¼ 2 þ dn dn
whs ðnÞ ¼
6 Enh rh2
n2 Js ðnÞ Nf s ð1Þ 2
ð43Þ
ð44Þ
5. PZT used as a sensor When the host beam undergoes mechanical deformation, the bonded PZT patch deforms and the produced electrical charge can be used as a sensing signal. To use the PZT patch as a sensor, we need to derive sensing equations for the smart beam under the action of the applied axial force (F0 ), the transverse force (P0 ) and the bending moment (M0 ) as shown in Fig. 4.
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5.1. Shear stress and peel stress for the applied forces in the host beam Under the action of the applied axial force or the bending moment, the expressions of the shear and peel stresses are the same in format as those under the action of the applied voltage except for the definitions of the following constants. (a) When an axial force (F0 ) is applied to the host beam, we have F0 and k ¼ I; II Enh rh Ga L (b) When a bending moment (M0 ) is applied to the host beam, we find Hnk ¼ ra en ;
Hmk ¼ Hqk ¼ 0
Hnk ¼ ra em ;
Hmk ¼
2rav em ; rh
where; en ¼
Hqk ¼ 0
where; em ¼
6M0 and k ¼ I; II Enh rh2 Ga L2
(c) When the host beam is subjected to a transverse shear force (P0 ) at the free end, we obtain 9 1 1 > > ðHnII þ HnI Þ ¼ ra aL eq ; ðHnII HnI Þ ¼ ra eq > > 2 2 > > > 1 2rav 1 2rav = ðHmII þ HmI Þ ¼ ðHmII HmI Þ ¼ aL e q ; eq 2 2 rh rh > > > > > > 1 2rav 1 > ; ðHqII þ HqI Þ ¼ ðHqII HqI Þ ¼ 0 eq ; 2 2 rh
ð45Þ
ð46Þ
ð47Þ
where, b aL ¼ ; L
eq ¼
6P0 Enh rh2 Ga L
ð48Þ
In this case, distributions of the shear stress and the peel stress are different from those of the applied axial force or bending moment. In additional to the anti-symmetric shear stress and symmetric peel stress components, there exist the symmetric shear stress and anti-symmetric peel stress components and are given by snsym ¼ A2 cosh b1 n þ A3 sinh b2 n sin b3 n þ A6 cosh b2 n cos b3 n þ A7 ð49Þ rnant ¼ B1 sinh b1 n þ B4 cosh b2 n sin b3 n þ B5 sinh b2 n cos b3 n However, it is worth noting that the sensing electric charge is produced only by the anti-symmetric shear stress and symmetric peel stress components. 5.2. Sensing equation Lee (1992) derived the following sensor equation: Z Z ou0 ov0 ou0 ov0 þ e36 qk ¼ e31 þ e32 þ þ eT33 E3 dx dy ox oy oy ox S ð12Þ Z Z o2 w o2 w o2 w e31 2 þ e32 2 þ 2e36 dx dy z0k ox oy oxoy S ð12Þ
ð50Þ
where, qk is the electric charge on layer k of the laminates, and z0k ¼ ðzk þ zk1 Þ=2 is the half thickness of that layer.
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Substituting the non-dimensional displacements of the beam into Eq. (50), we find Z 1 dun1 zk þ zk1 d2 wn1 qn ¼ e31 dn dn 2 dn2 1 where, qn ¼ q=L; q is the electric charge on the surface of PZT. By employing the constitutive relations of the PZT, we have Z 1 e31 qn ¼ Nn1 dn En1 r1 1
ð51Þ
ð52Þ
Substituting the exact solutions of the shear stress produced by the applied forces, we can obtain the following sensing equation: 2e31 dJs ð1Þ qn ¼ Nf s ð1Þ ð53Þ dn En1 r1 6. The host beam with two PZTs bonded symmetrically on both sides Crawley and de Luis (1987), and Im and Atluri (1989) investigated the active behaviors of the smart beam with the bonded PZT symmetrically. It is assumed that the poling direction of the PZT patch on the top surface is aligned with that of the PZT patch on the lower surface. When a pair of PZT patches is
Fig. 5. FBD for each element of the host beam with two PZT patches: (a) a smart beam with two symmetrically bonded PZT patches, (b) the PZT1 infinitesimal element, (c) the infinitesimal element of adhesive layer 1, (d) the infinitesimal element of the host beam, (e) the inifinitesimal element of adhesive layer 2 and (f) the PZT2 infinitesimal element.
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bonded to the top and bottom surfaces of the host beam symmetrically, as shown in Fig. 5, we can obtain the following non-dimensional equilibrium and constitutive equations. (a) The equilibrium equations are dNn1 þ sn1 ¼ 0; dn
dQn1 þ rn1 ¼ 0; dn
dNnh þ sn2 sn1 ¼ 0; dn dNn2 sn2 ¼ 0; dn
dMn1 1 þ r1 sn1 Qn1 ¼ 0 2 dn
dQnh þ rn2 rn1 ¼ 0; dn
9 > > > > > > > =
dMnh 1 þ rh ðsn2 þ sn1 Þ Qnh ¼ 0 > 2 dn > > > > > dMn2 1 > ; þ r1 sn2 Qn2 ¼ 0 2 dn
dQn2 rn2 ¼ 0; dn
ð54Þ
(b) By referring to Eqs. (6) and (7) and Fig. 5, the stress–displacement relations of the adhesive layers are 9 Rta dwnh dwn1 > > sn1 ¼ ra ðunh un1 Þ þ Rht þ ; rn1 ¼ rav ðwnh wn1 Þ > = 2 dn dn ð55Þ > Rta dwnh dwn2 > > Rht þ sn2 ¼ ra ðun2 unh Þ þ ; rn2 ¼ rav ðwn2 wnh Þ ; 2 dn dn (c) The constitutive equations of the host beam and PZT are 9 dun1 e31 V1 dunh dun2 e31 V2 > > Nn1 ¼ En1 r1 þ ; Nnh ¼ Enh rh ; Nn2 ¼ En1 r1 þ ; > = dn Ga L dn dx Ga L 1 d2 wn1 1 d2 wnh 1 d2 wn2 > > > Mn1 ¼ En1 r13 ; Mnh ¼ Enh rh3 ; Mn2 ¼ En1 r13 2 2 2 ; 12 12 12 dn dn dn
ð56Þ
The symbol definitions are the same as those of the host beam with PZT on its top surface, and i ¼ 1; 2; h, in which subscripts 1 and 2 refer to the top and lower PZT patches and subscript h refers to the host beam. Eqs. (54)–(56) can be transferred into two sets of the independent equations for extension and bending that can be solved separately. Case 1. [Host beam deformed in extension] When the host beam undergoes extension only, we have the following set of equations, in which, the following transforming parameters are used: 2Ns ¼ Nn1 þ Nn2 ; 2Qs ¼ Qn1 Qn2 ; 2us ¼ un1 þ un2 ; 2ws ¼ wn1 wn2 2ss ¼ sn1 sn2 ;
2rs ¼ rn1 þ rn2 ;
9 2Ms ¼ Mn1 Mn2 > =
2Vs ¼ V1 þ V2
> ;
ð57Þ
The equilibrium equations are dNs dQs þ ss ¼ 0; þ rs ¼ 0; dn dn dNnh 2ss ¼ 0 dn
9 dMs 1 þ r1 ss Qs ¼ 0 > = 2 dn > ;
ð58Þ
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The relations between the stresses/stress resultants and displacement are given by 9 dus e31 Vs 1 d2 ws dunh > > Ns ¼ En1 r1 þ ; Ms ¼ En1 r13 ; N ¼ E r = nh nh h 12 dn Ga L dn dn2 ð59Þ > Rta dws > ; ss ¼ ra ðunh us Þ þ ; rs ¼ rav ws 2 dn When both PZTs are used as actuators, extensional deformation can be achieved by applying a voltage (Vs ) of the same magnitude to both PZT patches so that the electric fields in both PZTs are in the same direction. Case 2. [Host beam deformed in bending] When the host beam undergoes the pure bending deformation, the following transforming parameters are used: 9 2Na ¼ Nn1 Nn2 ; 2Qa ¼ Qn1 þ Qn2 ; 2Ma ¼ Mn1 þ Mn2 = ð60Þ 2ua ¼ un1 un2 ; 2wa ¼ wn1 þ wn2 ; 2sa ¼ sn1 þ sn2 ; 2ra ¼ rn1 rn2 ; 2Va ¼ V1 V2 The equilibrium equations are 9 dNa dQa dMa 1 = þ sa ¼ 0; þ ra ¼ 0; þ r1 sa Qa ¼ 0 > 2 dn dn dn dQnh dMnh > ; 2ra ¼ 0; þ rh sa Qnh ¼ 0 dn dn
ð61Þ
and the stress/stress resultant–displacement relationships are given as
9 2 dua e31 Va 1 d2 wa 1 3 d wnh > > Na ¼ En1 r1 E þ ; Ma ¼ En1 r13 ; M ¼ r = nh nh h 12 12 dn G dn2 dn2 aL > Rta dwnh dwa > ; Rht þ s a ¼ r a ua þ ; ra ¼ rav ðwnh wa Þ 2 dn dn
ð62Þ
When both PZTs are used as actuators, bending deformation can be achieved by applying a voltage (Va ) of the same magnitude but opposite directions to both PZT patches so that the electric fields in both PZTs have the same magnitude but opposite directions. The equations for Case 1 can be transformed into the coupled ODE as shown in Eq. (8), in which 9 Rta 4 2 6Rta > > kn1 ¼ ks1 ¼ 2 þ ; kn2 ¼ ks2 ¼ = r1 En1 Enh Rht En1 r13 ð63Þ rav 12rav Rta > > kn3 ¼ ks3 ¼ ks2 ; kn4 ¼ ks4 ¼ ; ra En1 ra r14 Similarly, the coupled ODE as given in Eq. (8) can also be derived for Case 2, in which 9 Rta 4 6 6Rta 1 2 > > kn1 ¼ ka1 ¼ 2 þ ; kn2 ¼ ka2 ¼ 3 > En1 Enh R2ht = r1 En1 Enh Rht r1 ð64Þ > rav 12rav Rta 1 2 > > þ kn3 ¼ ka3 ¼ ka2 ; kn4 ¼ ka4 ¼ ; En1 Enh R3ht ra ra r14 Solutions to Cases 1 and 2 can also be expressed as Eqs. (15) and p (16). ffiffiffi When Enh R2ht En1 ¼ 0, for example, it is equivalent to that h ¼ 2t1 for the same elastic moduli of the host beam and PZT. In this case, ka2 ¼ ka3 ¼ 0 and the coupled ODEs become decoupled for Case 2: d 3 sa d4 ra 2 dsa ; ¼ b ¼ b4ua2 ra ua1 dn dn3 dn4 pffiffiffiffiffiffi pffiffiffiffiffiffi where, bua1 ¼ ka1 , bua2 ¼ 4 ka4 .
ð65Þ
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Solution to Eq. (65) is similar to that of Eq. (10) with the integration constants given in Appendix A. The polynomials for determining the eigenvalues can be obtained by replacing kni (i ¼ 1–4) with kri (r ¼ s, a; i ¼ 1–4) and solutions to the shear stress and peel stress are also similar to those of the host beam with the bonded PZT on the top surface. The boundary conditions can be expressed as d2 wsk 12Msk d3 wsk 6ssk 12Qsk ¼ ; ¼ ; 2 3 3 2 E r E r En1 r13 dn dn n1 1 n1 1 du N d2 wak 12Mak d3 wak 6ssk 12Qsk n ¼ 1 : ak ¼ ak þ eae ; ¼ ; ¼ ; 2 3 3 2 dn En1 r1 En1 r1 En1 r13 En1 r1 dn dn dunhk Nnhk d2 wnhk 12Mnhk d3 wnhk 12sak 12Qnhk ¼ ; ¼ ; ¼ dn Enh rh Enh rh2 Enh rh3 Enh rh3 dn2 dn3 dusk Nsk ¼ þ ese ; dn En1 r1
e31 Vs ese ¼ E1 t 1
9 > > > > > > > > =
e31 Va E1 t 1 > > > > > > > ; ðk ¼ I; IIÞ >
eae ¼
ð66Þ
in which, ese and eae are the electrically actuated strains. The boundary conditions expressed in terms of shear and peel stresses for Case 1 are: n ¼ 1 :
dss ¼ Hnsk ; dn
d2 rs ¼ Hmsk ; dn2
d3 rs þ ks3 ss ¼ Hqsk dn3
ðk ¼ I; IIÞ
ð67Þ
where, 9 Nsk Nnhk 6Rta Msk > > Hnsk ¼ ra þ ese þ > En1 r1 Enh rh En1 r13 = ðk ¼ I; IIÞ > 12rav Msk 12rav Qsk > > Hmsk ¼ ; Hqsk ¼ ; En1 r13 En1 r13
ð68Þ
The boundary conditions expressed in terms of shear and peel stresses for Case 2 are: n ¼ 1 :
dsa ¼ Hnak ; dn
d2 ra ¼ Hmak ; dn2
d3 ra þ ka3 ss ¼ Hqak dn3
ðk ¼ I; IIÞ
ð69Þ
where,
9 Nak Mnhk Mak > > Hnak ¼ ra þ eae þ 6Rta Rht þ > > = En1 r1 Enh rh3 En1 r13 ðk ¼ I; IIÞ > Mak Mnhk Qak Qnhk > > > Hmak ¼ 12rav ; Hqak ¼ 12rav ; En1 r13 Enh rh3 En1 r13 Enh rh3
ð70Þ
We now obtain the same forms of the shear stress, the peel stress and the boundary conditions in Cases 1 and 2 for the smart beam with the bonded PZTs symmetrically, and they are also the same as those obtained for the smart beam with the PZT patch on the top surface. Therefore, the developed solutions for the case with the PZT on the top surface can be readily employed. Having solved the ss ; rs and sa ; ra , we can obtain the shear and peel stresses in the adhesive layers: sn1 ¼ ss þ sa ;
sn2 ¼ sa ss
and rn1 ¼ rs þ ra ;
rn2 ¼ rs ra
ð71Þ
Using the similar procedures to those used in the host beam with one PZT on the top surface, the actuated internal forces, displacements and the sensing electric charges can be derived. Because the forms of the equilibrium equations, constitutive relations and boundary conditions are the same as those in the case with the PZT on the top surface, the related formulations can also be used directly.
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When the PZTs are used as actuators and the same voltages are applied to both PZTs with the same poling direction, the host beam will be deformed in extension and the pure bending occurs for the same value but opposite voltages. The phenomena are similar to those predicted by Crawley and de Luis (1987), but the values are different for flexible structures, which will be discussed in Part II.
7. Conclusion This paper takes into account the peel stress in the mathematical formulations for smart beams with PZT actuators and sensors. Exact solutions for both shear and peel stresses are developed in general and explicit form. The exact solutions are then used to develop analytical solutions for two cases: (a) a cantilever beam with one PZT patch bonded to one surface of the host beam; and (b) a cantilever beam with two PZT patches symmetrically-bonded both surfaces of the host beam. Exact formulations are obtained for the actuated force, bending moment and displacement as well as sensing equation. A mathematical formulation is also obtained for the conditions when the coupled peel and shear equations can be decoupled.
Acknowledgements The authors are grateful to the support of the Australian Research Council through a Large Grant Scheme (grant no. A10009074). Appendix A. Solutions to the integration constants Solutions to Eqs. (26) and (28) are as follows: 1 HnII þ HnI HmII þ HmI HqII HqI A1 ¼ bn bm þ bq D 2 2 2 1 A4 ¼ ½ðb cosh b2 cos b3 bn3 sinh b2 sin b3 ÞðHnII þ HnI Þ Dbq n2 ðbm2 cosh b2 cos b3 bm3 sinh b2 sin b3 ÞðHmII þ HmI Þ þ ðb12 cosh b2 cos b3 b13 sinh b2 sin b3 Þbq cosh b1 ðHqII HqI Þ 1 ½ ðbn2 sinh b2 sin b3 þ bn3 cosh b2 cos b3 ÞðHnII þ HnI Þ A5 ¼ Dbq
þðbm2 sinh b2 sin b3 þ bm3 cosh b2 cos b3 ÞðHmII þ HmI Þ ðb12 sinh b2 sin b3 þ b13 cosh b2 cos b3 Þbq cosh b1 ðHqII HqI Þ
9 > > > > > > > > > > > > > > > > > > =
> > > > > > > > > > > > > > > > > > ;
ðA:1Þ
where,
9 D ¼ bL1 bq sinh b1 þ bD cosh b1 > > > bn ¼ ðbK2 bL3 þ bK3 bL2 Þ sinh 2b2 þ ðbK2 bL2 bK3 bL3 Þ sin 2b3 > > > > > > bm ¼ ðb2 bL3 þ b3 bL2 Þ sinh 2b2 ðb2 bL2 b3 bL3 Þ sin 2b3 > > = bq ¼ ðb3 bK2 b2 bK3 Þðcosh 2b2 þ cos 2b3 Þ bD ¼ ðb12 bL3 þ b13 bL2 Þ sinh 2b2 þ ðb12 bL2 b13 bL3 Þ sin 2b3 > > > > > bn2 ¼ DbK2 bn b12 cosh b1 ; bn3 ¼ DbK3 bn b13 cosh b1 > > > > bm2 ¼ Db2 þ bm b12 cosh b1 ; bm3 ¼ Db3 þ bm b13 cosh b1 > > ; b12 ¼ bK1 b2 b1 bK2 ; b13 ¼ bK1 b3 b1 bK3
ðA:2Þ
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9 > > > > > > > > > > > > > > > > > > > =
K1 HnII þ HnI HmII þ HmI HqII HqI bn bm þ bq D 2 2 2 1 B3 ¼ f½ðK2 bn2 K3 bn3 Þ cosh b2 cos b3 ðK2 bn3 þ K3 bn2 Þ sinh b2 sin b3 ðHnII þ HnI Þ Dbq
B2 ¼
½ðK2 bm2 K3 bm3 Þ cosh b2 cos b3 ðK2 bm3 þ K3 bm2 Þ sinh b2 sin b3 ðHmII þ HmI Þ þ ½ðK2 b12 K3 b13 Þ cosh b2 cos b3 ðK2 b13 þ K3 b12 Þ sinh b2 sin b3 bq ðHqII HqI Þ
> > > > > > 1 > B6 ¼ f ½ðK3 bn2 þ K2 bn3 Þ cosh b2 cos b3 ðK3 bn3 K2 bn2 Þ sinh b2 sin b3 ðHnII þ HnI Þ > > > Dbq > > > > > > þ½ðK3 bm2 þ K2 bm3 Þ cosh b2 cos b3 ðK3 bm3 K2 bm2 Þ sinh b2 sin b3 ðHmII þ HmI Þ > > > ; ½ðK3 b12 þ K2 b13 Þ cosh b2 cos b3 ðK3 b13 K2 b12 Þ sinh b2 sin b3 bq ðHqII HqI Þ 1 HnII HnI HmII HmI HqII þ HqI kn3 ðNn1II Nn1I Þ A2 ¼ bm0 þ bq0 bn0 D0 2 2 2 2 1 A3 ¼ fðbn02 sinh b2 cos b3 bn03 cosh b2 sin b3 ÞðHnII HnI Þ D0 bq0 ðbm02 sinh b2 cos b3 bm03 cosh b2 sin b3 ÞðHmII HmI Þ
9 > > > > > > > > > > > > > > > > > > > =
þ ðb12 sinh b2 cos b3 b13 cosh b2 sin b3 Þbq0 sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ > > > > > > 1 > > A6 ¼ f½ðbn02 cosh b2 sin b3 þ bn03 sinh b2 cos b3 ÞðHnII HnI Þ > > D0 bq0 > > > > > > þðbm02 cosh b2 sin b3 þ bm03 sinh b2 cos b3 ÞðHmII HmI Þ > > > ; ðb12 cosh b2 sin b3 þ b13 sinh b2 cos b3 Þbq0 sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ A7 ¼
NnI1I Nn1I HnII HnI HmII HmI þ ðbr1 bn0 sinh b1 þ brn Þ þ ðbr1 bm0 sinh b1 brm Þ 2 2D0 2D0 HqII þ HqI kn3 ðNn1II Nn1I Þ ðbr1 brq Þbq0 sinh b1 2D0 2D0
ðA:3Þ
ðA:4Þ
ðA:5Þ
where, 9 > > > > > bn0 ¼ ½bK2 bL3 þ bK3 ðbl21 sin2 b3 þ bl22 cos2 b3 Þ sinh 2b2 > > > > 2 2 > þ½bK3 bL3 þ bK2 ðbl21 sinh b2 bl22 cosh b2 Þ sin 2b3 > > > > 2 > 2 > bm0 ¼ ½b2 bL3 þ b3 ðbl21 sin b3 þ bl22 cos b3 Þ sinh 2b2 > > > 2 2 > þ½b2 ðbl22 cosh b2 bl21 sinh b2 Þ b3 bL3 sin 2b3 > > > > = b ¼ ðb b b b Þðcosh 2b cos 2b Þ D0 ¼ bL11 bq0 cosh b1 þ bD0 sinh b1
q0
3 K2
2 K3
2
3
> > > > > 2 2 > þ½b13 bL3 þ b12 ðbl21 sinh b2 bl22 cosh b2 Þ sin 2b3 > > > > > bn02 ¼ D0 bK2 bn0 b12 sinh b1 ; bn3 ¼ D0 bK3 bn0 b13 sinh b1 > > > > > bm02 ¼ D0 b2 þ bm0 b12 sinh b1 ; bm03 ¼ D0 b3 þ bm0 b13 sinh b1 > > > > > > b3 bn02 þ b2 bn03 > > brn ¼ 2 ; 2 ðb2 þ b3 Þðb3 bK2 b2 bK3 Þ 2
bD0 ¼ ½b12 bL3 þ b13 ðbl21 sin b3 þ bl22 cos2 b3 Þ sinh 2b2
ðA:6Þ
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K1 HnII HnI HmII HmI HqII þ HqI kn3 ðNn1II Nn1I Þ bn0 bm0 þ bq0 2 D 2 2 2 1 B4 ¼ f½ðK2 bn02 K3 bn03 Þ sinh2 cos b3 ðK2 bn03 þ K3 bn02 Þ cosh b2 sin b3 ðHnII HnI Þ D0 bq0
B1 ¼
½ðK2 bm02 K3 bm03 Þ sinh b2 cos b3 ðK2 bm03 þ K3 bm02 Þ cosh b2 sin b3 ðHmII HmI Þ
9 > > > > > > > > > > > > > > > > > > > > > > > =
þ½ðK2 b12 K3 b13 Þ sinh b2 cos b3 ðK2 b13 þ K3 b12 Þ cosh b2 sin b3 bq0 sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ > > > > > 1 > > B5 ¼ ½ðK b þ K b Þ sinh b cos b ðK b K b Þ cosh b sin b ðH H Þ f 3 n02 2 n03 3 n03 2 n02 nII nI > 2 3 2 3 > > D0 bq0 > > > > > þ½ðK3 bm02 þ K2 bm03 Þ sinh b2 cos b3 ðK3 bmo3 K2 bmo2 Þ cosh b2 sin b3 ðHmII HmI Þ > > > > > ½ðK3 b12 þ K2 b13 Þ sinh b2 cos b3 ðK3 b13 K2 b12 Þ cosh b2 sin b3 > > > ; bq sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ ðA:7Þ If the substructures have the same material properties and the geometric sizes, solutions to the integration constants are 9 HnII þ HnI > A1 ¼ > > > 2bsn1 cosh bsn1 > > > ðsinh be cos be þ cosh be sin be Þbe ðHmII þ HmI Þ sinh be sin be ðHqII HqI Þ = B1 ¼ ðA:8Þ 2b3e ðsinh 2be þ sin 2be Þ > > > > ðsinh be cos be cosh be sin be Þbe ðHmII þ HmI Þ cosh be cos be ðHqII HqI Þ > > > ; B4 ¼ 2b3e ðsinh 2be þ sin 2be Þ A2 ¼
HnII HnI ; 2bsn1 sinh bsn1
A3 ¼
Nn1II Nn1I HnII HnI 2 2b2sn1
9 ðsinh be sin be þ cosh be cos be Þbe ðHmII HmI Þ cosh be sin be ðHqII þ HqI Þ > > > > B2 ¼ = 2b3e ðsinh 2be sin 2be Þ ðcosh be cos be sinh be sin be Þbe ðHmII HmI Þ sinh be cos be ðHqII þ HqI Þ > > > B3 ¼ > ; 2b3e ðsinh 2be sin 2be Þ
ðA:9Þ
ðA:10Þ
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Dokumaci, E., 1987. An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry. Journal of Sound and Vibration 119 (3), 443–449. Goland, M., Reissner, E., 1944. The stresses in cemented joints. Journal of Applied Mechanics A, 17–27. Holland, R., EerNisse, E.P., 1969. Design of Resonant Piezoelectric Devices. MIT Press, Cambridge, MA. Humi, M., Miller, W., 1988. Second Course in Ordinary Differential Equations for Scientists and Engineers. Springer-Verlag, Berlin. Im, S., Atluri, S.N., 1989. Effects of a piezo-actuator on a finitely deformed beam subjected to general loading. AIAA Journal 27 (12), 549–564. Kim, S.J., Jones, J.D., 1996. Effects of piezo-actuator delamination on the performance of active noise and vibration control system. Journal of Intelligent Material Systems and Structures 7 (6), 668–676. Lee, C.-K., 1992. Piezoelectric laminates: Theory and experiments for distributed sensors and actuators. In: Tzou, H.S., Anderson, G.L. (Eds.), Intelligent Structural Systems. Kluwer Academic, Dordrecht, pp. 75–135. Post, D., Wood, J.D., John, D., 1988. Deformations and strains in a thick adherend lap joint. In: Johnson, W.S. (Ed.), Adhesively Bonded Joints: Testing, Analysis and Design. ASTM Publication, pp. 107–118. Robbins, D.H., Reddy, J.N., 1991. Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Computers and Structures 41 (2), 265–279. Seeley, C.E., Chattopadhyay, A., 1998. Experimental investigation of composite beams with piezoelectric actuation and debonding. Smart Materials and Structures 7, 502–511. Seeley, C.E., Chattopadhyay, A., 1999. Modeling of adaptive composites including debonding. International Journal of Solids and Structures 36, 1823–1843. Shi, G., Atluri, S.N., 1990. Active control of nonlinear dynamic response of space-frames using piezo-electric actuators. Computers and Structures 34 (4), 1801–1807. Suhir, E., 1986. Stresses in bi-metal thermostats. Journal of Applied Mechanics 53, 657–660. Suhir, E., 2000. Adhesively bonded assemblies with identical nondeformable adherends and ‘piecewise continuous’ adhesive layer: Predicted thermal stresses in the adhesive. International Journal of Solids and Structures 37, 2229–2252. Sun, D., Tong, L., Atluri, S.N., 2001. Effects of piezoelectric sensor/actuator debonding on vibration control of smart beams. International Journal of Solids and Structures 38, 9033–9051. Tanaka, M., Bercin, A.N., 1999. Free vibration for uniform beams of nonsymmetrical cross-section using mathematica. Computers and Structures 71, 1–8. Timoshenko, S.P., Gere, J.M., 1972. Mechanics of Materials. Van Nostrand-Reinhold, New York. Tong, L., Steven, G.P., 1999. Analysis and Design of Structural Bonded Joints. Kluwer Academic, Boston. Tong, L., Sun, D., Atluri, S.N., 2001. Sensing and actuating behaviors of piezoelectric layer in smart beams with debonding. Smart Materials and Structures 10 (4), 713–723. Uspensky, J.V., 1948. Theory of Equations. McGraw-Hill, New York. Wang, B.T., Rogers, C.A., 1991. Modeling of finite-length spatially-distributed induced strain actuators for laminate beams and plates. Journal of Intelligent System and Structures 2, 38–58. Wang, X.D., Meguid, S.A., 2000. On the electroelastic behavior of a thin piezoelectric actuator attached to an infinite host structure. International Journal of Solids and Structures 37, 3231–3251.
Exact static solutions to piezoelectric smart beams including peel stresses I: Theoretical formulation Quantian Luo, Liyong Tong
*
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia Received 14 November 2001; received in revised form 20 May 2002
Abstract This paper presents exact static solutions to smart beams with perfectly bonded or partially debonded piezoelectric (PZT) actuators and sensors including peel stresses. When a PZT patch is bonded on the surface of a beam, both shear and peel stresses exist in the adhesive between the PZT patch and the host beam. In Part I, non-dimensional differential governing equations for infinitesimal elements of the PZT patch and the host beam are formulated firstly, and then the ordinary differential equation (ODE) of the coupled shear and peel stresses are completely solved analytically. The exact solutions are applicable to smart beams with PZT actuators and sensors. When the PZT patches are used as actuators, the solutions give the formulations for the energy transferring shear and peel stresses in the adhesive actuated by the applied voltages, and for the actuated stress resultants and displacements in the host beam. When the PZT patches are used as sensors, the solutions give the shear and peel stresses in the adhesive caused by the loadings, and also the sensing electric charges. When PZT patches are not perfectly bonded to the host beam, i.e., partially debonded, the developed solutions can be tailored to the case of edge debondings, i.e., PZTs with the shortened length. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectric actuator and sensor; Peel stress; Coupled differential equation
1. Introduction A smart structure with piezoelectric (PZT) actuators or sensors has drawn a considerable interest in structural performance control due to the coupled mechanical and electrical properties of PZT materials. PZTs are normally bonded to or embedded in the host structure in structural applications such as vibration and shape control. Crawley and de Luis (1987) developed a theoretical framework for modeling extensional and bending deformations of a beam with PZT actuators, which has been frequently cited and widely used
*
Corresponding author. Tel.: +61-2-9351-6949; fax: +61-2-9351-4841. E-mail address: [email protected] (L. Tong).
0020-7683/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 0 2 ) 0 0 3 8 3 - 9
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as the theoretical fundamentals. Their theory assumed that PZT patches are perfectly bonded to the host structures and only shear stresses exist in the adhesive layer. Using the similar assumptions, Crawley and Lazarus (1991) applied the theory to plates with perfectly bonded PZTs. Im and Atluri (1989) formulated governing equations for a beam column linked with piezo-actuators subjected to general loading, in which the peel stresses were considered. However, they believed that the peel stresses could be ignored for most cases and the peel stresses were neglected in their subsequent analysis. Shi and Atluri (1990) utilized these results for active control of space frame vibrations. They also assumed that PZT patches were perfectly bonded to the host structures. Robbins and Reddy (1991) modeled the adhesive layer as a beam using a layer-wise displacement theory. Although a beam model for the adhesive layer could be used to study the peel effects, Goland and Reissner (1944) showed that the thin flexible joint layer would not bear bending moments and behave like a spring with the shear and peel stiffness. Anderson et al. (1977) described the analysis and tests for this adhesive behavior. Post et al. (1988) and Dillard et al. (1988) testified this characteristic of the adhesive layer by photo-elastic and mechanical experiments respectively. Tong and Steven (1999) gave systematic description for mechanical behaviors of structural loaded joints. All of these works modeled the adhesive layer as the spring system with the shear and peel stiffness. Suhir (1986, 2000) also employed this adhesive model for analyzing thermal stresses in the adhesive. The classic theory of Goland and Reissner (1944) has been shown to predict the reasonable stress distributions in the thin adhesive layer. Kim and Jones (1996) conducted an analytical and experimental investigation to identify the effects of delamination on the smart beam with PZT. Seeley and Chattopadhyay (1998) also performed the experimental investigation of composite beams with PZT actuation and debonding, and then they (Seeley and Chattopadhyay, 1999) further studied debonding effects of the smart beams using a refined higher order finite element analysis. Wang and Meguid (2000) utilized the solutions of a whole plane and a half plane subjected to a concentrated horizontal force for analyzing debonding effects of the embedded and bonded PZT patch. These models did not model the adhesive layer and thus may not be used to investigate the peel effects. Tong et al. (2001) and Sun et al. (2001) modeled the adhesive layer as a shear and peel spring system and studied the effects of debondings, stresses in the adhesive layer and stress resultants for smart beams. In their analysis, the multi-segment shooting method was used to solve the governing equations. In the present study, based on the classic theory in bonding joints (Tong and Steven, 1999), a theoretical model for a PZT composite beam including adhesive layers is developed. In this model, PZT patches and the host beam are modeled as Euler–Bernoulli beams, and the adhesive layer is modeled as a continuous spring with the shear and peel stiffness. Using this model, differential equations of the equilibrium are formulated. Incorporating the constitutive relations of PZT, the adhesive layer, and the host beam with the equilibrium equations, we can obtain the differential equations of the coupled shear and peel stresses. The governing equations can be completely solved analytically, and the shear and peel stresses in the adhesive are exactly obtained. Based on analytical solutions of the shear and peel stresses, we can derive the actuated internal forces and displacements of the host beam as well as the sensing electric charge of the PZTs. In this analytical model, the edge debondings are also studied by simply considering a shortened PZT length.
2. Basic differential equation Consider a PZT patch bonded to the host beam on the top surface, as shown in Fig. 1, the free body diagrams for the infinitesimal elements of the PZT, adhesive layer and host beam are depicted in Fig. 2. In the light of static equilibrium conditions, we can perform the following formulation, in which the unit width of the whole beam with the rectangular cross section.
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Fig. 1. PZT composite beam.
Fig. 2. Stresses and forces on the infinitesimal elements of the substructures: (a) the PZT infinitesimal element, (b) the adhesive layer element and (c) the host beam element.
The following parameters are introduced for the non-dimensional formulations: 8 Ni Qi Mi Ei s r ui > > ; Qni ¼ ; Mni ¼ ; Eni ¼ ; sn ¼ ; rn ¼ ; uni ¼ ; < Nni ¼ 2 Ga Ga Ga L Ga L Ga L Ga L wi ti x L Ea t1 h > > : wni ¼ ri ¼ ; n ¼ ; ra ¼ ; rav ¼ ra ; Rta ¼ ; Rht ¼ ði ¼ 1; hÞ L ta t1 L L Ga ta
ð1Þ
The subscripts 1, h, and a in the above and all subsequent equations refer to the PZT patch, the host beam and the adhesive layer. In all above equations, Ei , and ti (ti ¼ t1 , h) are the elastic moduli and the thickness of the sub-structure; ui and wi (i ¼ 1; h) are the extensional and flexural displacements; e31 and V1 are a PZT coupling constant of the PZT and the applied voltage; Ea and Ga are the Young’s and shear moduli of the adhesive layer. The remaining mathematical symbols are defined in Fig. 2.
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As shown in Fig. 2, the non-dimensional ordinary differential equations (ODEs) of the equilibrium are: dNn1 þ sn ¼ 0; dn
dQn1 þ rn ¼ 0; dn
dMn1 1 þ r1 sn Qn1 ¼ 0 2 dn
ð2Þ
dNnh sn ¼ 0; dn
dQnh rn ¼ 0; dn
dMnh 1 þ rh sn Qnh ¼ 0 2 dn
ð3Þ
By employing the mechanical–electrical relations of PZT (Holland and EerNisse, 1969; Chee et al., 1998) and the Euler–Bernoulli beam theory (Timoshenko and Gere, 1972), we can obtain the following constitutive relations for the PZT and the host beam: Nn1 ¼ En1 r1
dun1 e31 V1 þ ; dn Ga L
Nnh ¼ Enh rh
dunh ; dn
Mn1 ¼
Mnh ¼
1 d2 wn1 En1 r13 12 dn2
ð4Þ
1 d2 wnh Enh rh3 12 dn2
ð5Þ
Referring to Fig. 2 and the definition given by Goland and Reissner (1944), we have the constitutive relations for the adhesive layer: Rta dwnh dwn1 Rht sn ¼ ra ðunh un1 Þ þ þ ð6Þ 2 dn dn rn ¼ rav ðwnh wn1 Þ
ð7Þ
Differentiating Eqs. (6) and (7), into which Eqs. (2)–(5) are substituted, we can obtain the following coupled differential equation: d 3 sn dsn þ kn2 rn ; ¼ kn1 3 dn dn
d4 rn dsn kn4 rn ¼ kn3 4 dn dn
ð8Þ
where, kn1 ¼
4Rta r12
rav kn3 ¼ kn2 ; ra
1 1 þ ; En1 Enh Rht 12rav Rta kn4 ¼ ra r14
kn2 ¼
6Rta r13
1 1 En1 Enh R2ht
1 1 þ En1 Enh R3ht
9 > > > = > > > ;
ð9Þ
When Enh R2ht En1 ¼ 0, kn2 and kn3 in Eq. (8) become zero, and thus governing equation (8) for the shear and peel stresses becomes decoupled and can be simplified as d 3 sn dsn ; ¼ b2uc1 3 dn dn
d4 rn ¼ b4uc2 rn dn4
ð10Þ
in which, b2uc1 ¼ kn1 ;
b4uc2 ¼ kn4
ð11Þ
The decoupling condition Enh R2ht En1 ¼ 0 can be rewritten as Eh h2 =E1 t12 ¼ 1. Evidently, when both the PZT and host beam have the same Young’s moduli and thickness, the decoupling condition is satisfied. This case was referred to as the balanced single-lap joints by Goland and Reissner (1944) and Tong and Steven (1999), for which the decoupled equation (8) was solved for the selected loadings and boundary conditions.
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3. Solution to the coupled equation (8) To solve the coupled governing equation (8), let’s assume sn ¼ A ebn and rn ¼ B ebn and then substitute into Eq. (8), we have ðb3 kn1 bÞA kn2 B ¼ 0;
kn3 bA þ ðb4 þ kn4 ÞB ¼ 0
ð12Þ
The condition of non-trivial solutions to the shear and peel stresses requires ½b6 kn1 b4 þ kn4 b2 ðkn1 kn4 kn2 kn3 Þb ¼ 0
ð13Þ
which becomes k3 kn1 k2 þ kn4 k ðkn1 kn4 kn2 kn3 Þ ¼ 0;
where; k ¼ b2 ; or b ¼ 0
ð14Þ
Roots of the above polynomial can be analytically solved (Uspensky, 1948), and there always exist one real number root and two conjugate complex number roots for the most combinations of typical material properties and geometric configurations. Therefore, we can express solutions to the shear and peel stresses as sn ¼ A1 sinh b1 n þ A2 cosh b1 n þ ðA3 sinh b2 n þ A4 cosh b2 nÞ sin b3 n þ ðA5 sinh b2 n þ A6 cosh b2 nÞ cos b3 n þ A7 rn ¼ B1 sinh b1 n þ B2 cosh b1 n þ ðB3 sinh b2 n þ B4 cosh b2 nÞ sin b3 n þ ðB5 sinh b2 n þ B6 cosh b2 nÞ cos b3 n þ B7
ð15Þ
ð16Þ
In which Ai and Bi (i ¼ 1–7) are the integration constants to be determined using the boundary conditions; b1 , b2 , and b3 are determined by solving Eq. (13). Combining both equations in Eq. (8), by eliminating the shear stress, we have an ODE of six-order for the peel stress. Substituting Eq. (16) into the six-order ODE and noting Eq. (13), we find B7 ¼ 0. The first equilibrium equation shown in Eq. (2) defines a supplementary equation for the shear stress: Z 1 sn ðnÞ dn ¼ Nn1II Nn1I ð17Þ 1
where, Nn1II and Nn1I are the non-dimensionalized axial forces applied at the PZT edges (see Fig. 3). Fig. 3 illustrates that one PZT patch is bonded on the top surface of the host beam. In Fig. 3, N1k , M1k and Q1k (k ¼ I; II) are the applied forces at the edges of the PZT or the internal forces at the debonding edges; Nhk ; Mhk and Qhk (k ¼ I; II) are the internal forces at cross-sections I–I and II–II of the host beam, which can be solved by the equilibrium conditions for the statically-determined structures.
Fig. 3. Boundary conditions of the host beam with the partially bonded PZT.
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Referring to Fig. 3, we directly obtain the non-dimensional forces in the boundaries using Eq. (1), and then, substituting the non-dimensional forces into the constitutive and equilibrium equations of the PZT patch and the host beam, we can solve the displacements at n ¼ 1: 9 dun1k Nn1k d2 wn1k 12Mn1k d3 wn1k 6snk 12Qn1k > > ¼ þ ee ; ¼ ; ¼ > dn En1 r1 En1 r12 En1 r13 En1 r13 = dn2 dn3 ðk ¼ I; IIÞ ð18Þ > dunhk Nnhk d2 wnhk 12Mnhk d3 wnhk 6snk 12Qnhk > > ; ¼ ; ¼ ; ¼ dn Enh rh2 En2 r22 En2 rh3 En2 rh3 dn2 dn3 In which, ee ð¼ e31 V1 =E1 t1 Þ is the electrically actuated strain. Substituting Eq. (18) into the constitutive relations of the adhesive, we obtain the following boundary conditions for solving the governing equation (8): 9 dsn > > ¼ Hnk > > dn > > 2 > = d rn ¼ H mk ðk ¼ I; IIÞ ð19Þ n ¼ 1 : dn2 > > 3 > > d rn > þ kn3 sn ¼ Hqk > > ; dn3 where,
9 Nn1k Nnhk Mn1k Mnhk > > Hnk ¼ ra þ ee þ Rht þ 6Rta > > En1 r1 Enh rh En1 r13 Enh rh3 > > > = Mnhk Mn1k Hmk ¼ 12rav 3 3 > Enh rh En1 r1 > > > > > Qnhk Qn1k > ; Hqk ¼ 12rav 3 3 Enh rh En1 r1
ðk ¼ I; IIÞ
ð20Þ
Solutions to Eqs. (15)–(17) with the boundary conditions defined in Eq. (19) cannot be directly solved, as 12 integration constants cannot be directly determined by six equations of the boundary conditions. This boundary value problem (BVP) of ODE can be solved using the method of elimination. This technique attempts to solve a given system of coupled differential equations by eliminating the coupled items and then reducing a system to the uncoupled higher order equations (Humi and Miller, 1988). For the ODE with constant coefficients in the present study, we can also obtain Eqs. (15)–(17), and then combine the governing equation (8) and the boundary conditions shown in Eq. (19) to obtain the higher order equations of the boundary conditions, which can be used to determine the integration constants. Alternatively, we may substitute Eqs. (15) and (16) into the first formulation or the second one of Eq. (8). In doing so, we can obtain six relations between Ai and Bi , and thus the integration constants can be uniquely solved. Dokumaci (1987) obtained an exact solution for the coupled bending and torsions of uniform beams having single cross-sectional symmetry using this method, which was the first exact solution in real form according to his literature search. Employing this method, Banerjee (1989) derived the coupled bending and torsion dynamic stiffness matrix for beam elements. Using Mathematica, Tanaka and Bercin (1999) solved the free vibration for uniform beams of non-symmetrical cross-section utilizing this method. The above two methods have been used for solving this BVP and the same results are obtained. Due to the space limit, only the solution procedure using the second one is presented here. Substituting Eqs. (15) and (16) into the first formulation of Eq. (8), we find B2 ¼ K1 A1 ;
B3 ¼ K2 A4 þ K3 A5 ;
B6 ¼ K3 A4 þ K2 A5
ð21Þ
B1 ¼ K 1 A2 ;
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B4 ¼ K2 A3 þ K3 A6 ;
ð22Þ
B5 ¼ K3 A3 þ K2 A6
where, K1 ¼
b1 ðb21 kn1 Þ ; kn2
K2 ¼
b2 ½ðb22 3b23 Þ kn1 ; kn2
K3 ¼
b3 ½ðb23 3b22 Þ þ kn1 kn2
ð23Þ
Substituting Eqs. (15) and (16) into Eq. (19), and considering Eq. (17), we can obtain Eqs. (24) and (25): 9 A1 b1 cosh b1 þ A4 ðb2 sinh b2 sin b3 þ b3 cosh b2 cos b3 Þ > > > HnII þ HnI > > > þA5 ðb2 cosh b2 cos b3 b3 sinh b2 sin b3 Þ ¼ > > 2 > > 2 2 2 > > B2 b1 cosh b1 þ B3 ½ðb2 b3 Þ sinh b2 sin b3 þ 2b2 b3 cosh b2 cos b3 > = H þ H mII mI 2 2 ð24Þ þB6 ½ðb2 b3 Þ cosh b2 cos b3 2b2 b3 sinh b2 sin b3 ¼ > 2 > > 3 2 2 2 2 > B2 b1 sinh b1 þ B3 ½ðb2 ðb2 3b3 Þ cosh b2 sin b3 b3 ðb3 3b2 Þ sinh b2 cos b3 Þ > > > > þB6 ½ðb2 ðb22 3b23 Þ sinh b2 cos b3 þ b3 ðb23 3b22 Þ cosh b2 sin b3 Þ > > > > HqII HqI > ; þkn3 ðA1 sinh b1 þ A4 cosh b2 sin b3 þ A5 sinh b2 cos b3 Þ ¼ 2 9 A2 b1 sinh b1 þ A3 ðb2 cosh b2 sin b3 þ b3 sinh b2 cos b3 Þ > > > HnII HnI > > þA6 ðb2 sinh b2 cos b3 b3 cosh b2 sin b3 Þ ¼ > > 2 > > 2 2 2 > > B1 b1 sinh b1 þ B4 ½ðb2 b3 Þ cosh b2 sin b3 þ 2b2 b3 sinh b2 cos b3 > > > H H > mII mI 2 2 > > þB5 ½ðb2 b3 Þ sinh b2 cos b3 2b2 b3 cosh b2 sin b3 ¼ > > 2 > > 3 2 2 2 2 B1 b1 cosh b1 þ B4 ½ðb2 ðb2 3b3 Þ sinh b2 sin b3 b3 ðb3 3b2 Þ cosh b2 cos b3 Þ = ð25Þ þB5 ½ðb2 ðb22 3b23 Þ cosh b2 cos b3 þ b3 ðb23 3b22 Þ sinh b2 sin b3 Þ > > > HqII þ HqI > > þkn3 ðA2 cosh b1 þ A3 sinh b2 sin b3 þ A6 cosh b2 cos b3 þ A7 Þ ¼ > > > 2 > > > 1 1 > > A2 sinh b1 þ A3 2 ðb2 cosh b2 sin b3 b3 sinh b2 cos b3 Þ > 2 > b1 > b2 þ b 3 > > > 1 Nn1II Nn1I > > ; þA6 2 ðb sinh b cos b þ b cosh b sin b Þ þ A ¼ 7 2 2 3 3 2 3 2 2 b2 þ b 3 Substituting Eq. (21) into (24), we can obtain a group of the linear algebraic equations: 9 A1 b1 cosh b1 þ A4 ðb2 sinh b2 sin b3 þ b3 cosh b2 cos b3 Þ > > > HnII þ HnI > > > þA5 ðb2 cosh b2 cos b3 b3 sinh b2 sin b3 Þ ¼ > > 2 > > > A1 bK1 cosh b1 þ A4 ðbK2 sinh b2 sin b3 þ bK3 cosh b2 cos b3 Þ > > = HmII þ HmI þA5 ðbK2 cosh b2 cos b3 bK3 sinh b2 sin b3 Þ ¼ > 2 > > > A1 bL1 sinh b1 þ A4 ðbL2 cosh b2 sin b3 bL3 sinh b2 cos b3 Þ > > > > HqII HqI > > > > þA5 ðbL2 sinh b2 cos b3 þ bL3 cosh b2 sin b3 Þ ¼ > ; 2
ð26Þ
where, 9 bK1 ¼ K1 b21 ; bK2 ¼ K2 ðb22 b23 Þ þ 2K3 b2 b3 ; bK3 ¼ 2K2 b2 b3 K3 ðb22 b23 Þ = bL1 ¼ K1 b31 þ kn3 ; bL2 ¼ K2 b2 ðb22 3b23 Þ K3 b3 ðb23 3b22 Þ þ kn3 ; bL3 ¼ K3 b2 ðb22 3b23 Þ þ K2 b3 ðb23 3b22 Þ
ð27Þ
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The integration constants A1 , A4 , A5 can be solved from Eq. (26), and then B2 , B3 , B6 are solved from Eq. (21). Similarly, substituting Eq. (22) into (25) and eliminating A7 , we can obtain the other group of the linear algebraic equations: 9 A2 b1 sinh b1 þ A3 ðb2 cosh b2 sin b3 þ b3 sinh b2 cos b3 Þ > > > HnII HnI > > þA6 ðb2 sinh b2 cos b3 b3 cosh b2 sin b3 Þ ¼ > > > 2 > > = A2 bK1 sinh b1 þ A3 ðbK2 cosh b2 sin b3 þ bK3 sinh b2 cos b3 Þ HmII HmI ð28Þ þA6 ðbK2 sinh b2 cos b3 bK3 cosh b2 sin b3 Þ ¼ > > > 2 > > > A2 bL11 cosh b1 þ A3 ðbL21 sinh b2 sin b3 bL3 cosh b2 cos b3 Þ > > HqII þ HqI kn3 ðNn1II Nn1I Þ > > ; þA6 ðbL22 cosh b2 cos b3 þ bL3 sinh b2 sin b3 Þ ¼ 2 2 where, bL11 ¼ K1 b31 þ kn3
kn3 ðb2 cthb2 b3 ctanb3 Þ þ b23 > > > > kn3 > 2 2 2 2 > ¼ K2 b2 ðb2 3b3 Þ K3 b3 ðb3 3b2 Þ þ kn3 2 ðb thb þ b tan b Þ ; 2 2 3 3 2 b2 þ b3
bL21 ¼ K2 b2 ðb22 3b23 Þ K3 b3 ðb23 3b22 Þ þ kn3 bL22
9 > > > > > > =
kn3 thb1 b1
b22
ð29Þ
The integration constants A2 ; A3 and A6 can be solved from Eq. (28), and then B1 ; B4 ; B5 are solved from Eq. (22) and A7 is solved by the last formulation of Eq. (25). Eqs. (26) and (28) are the linear algebraic equations that can be easily solved. The explicit expressions of all integration constants are given in Appendix A. The integration constants of the decoupled equation (10) for the present boundary conditions are also explicitly given in Appendix A.
4. PZT used as an actuator On the basis of the linear theory, the superposition principle can be applied to the exerted voltage and forces. To investigate performance of a PZT actuator, we only study the mechanical behaviors of the smart beam under action of the applied voltage in PZT. In this case, the boundary conditions become Hnk ¼ ra ee ;
Hmk ¼ 0
and
Hqk ¼ 0;
ðk ¼ I; IIÞ
ð30Þ
4.1. Shear and peel stresses If there are no interior debondings in the adhesive layer, we can find from Eqs. (20) and (25) that, integration constants A2 ; A3 ; A6 ; A7 ; B1 ; B4 , and B5 are equal to zero. In this case, only Eq. (26) needs to be solved. The solutions of the shear and peel stresses are sn ¼ A1 sinh b1 n þ A4 cosh b2 n sin b3 n þ A5 sinh b2 n cos b3 n
ð31Þ
rn ¼ B2 cosh b1 n þ B3 sinh b2 n sin b3 n þ B6 cosh b2 n cos b3 n
ð32Þ
In which, integration constants A1 ; A4 ; A5 are determined from Eq. (26), as given in Appendix A; B2 ; B3 ; B6 are obtained from Eq. (21). Having solved the shear and peel stresses, the actuated internal forces, displacements in the host beam can be obtained. Solutions to the decoupled equation (10) can be simplified as
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sn ¼ A1 sinh buc1 n rn ¼ B1 sinh be n sin be n þ B4 cosh be n cos be n;
9 =
pffiffiffi 2 b ; where; be ¼ 2 uc2
4685
ð33Þ
4.2. Internal forces Substituting the shear and peel stresses into differential forms of the equilibrium equations, we can derive the actuated internal forces in the host beam. The results are: Nnh ¼ Nf s ðnÞ Nf s ð1Þ
ð34Þ
Qnh ¼ ½B2 br1 sinh b1 n þ B3 br2 ðb2 cosh b2 n sin b3 n b3 sinh b2 n cos b3 nÞ þ B6 br2 ðb2 sinh b2 n cos b3 n þ b3 cosh b2 n sin b3 nÞ 1 Mnh ¼ rh ½Nf s ð1Þ Nf s ðnÞ þ ½Mf r ðnÞ Mf r ð1Þ 2
ð35Þ ð36Þ
where, 9 > > > > > þA5 br2 ðb2 cosh b2 n cos b3 n þ b3 sinh b2 n sin b3 nÞ > > > > 2 2 2 2 Mf r ðnÞ ¼ B2 br1 cosh b1 n þ B3 br2 ½ðb2 b3 Þ sinh b2 n sin b3 n 2b2 b3 cosh b2 n cos b3 n = > > þB6 b2r2 ½ðb22 b23 Þ cosh b2 n cos b3 n þ 2b2 b3 sinh b2 n sin b3 n > > > > > 1 1 > > br1 ¼ ; br2 ¼ 2 ; 2 b1 b2 þ b 3 Nf s ðnÞ ¼ A1 br1 cosh b1 n þ A4 br2 ðb2 sinh b2 n sin b3 n b3 cosh b2 n cos b3 nÞ
ð37Þ
The average axial force and bending moment are given by 9 dJs ð1Þ dJs ð0Þ > > Naver ¼ 0 Nnh dn ¼ Nf s ð1Þ > = dn dn R1 > 1 dJs ð1Þ dJs ð0Þ dJr ð1Þ dJr ð0Þ > þ Mf r ð1Þ > Maver ¼ 0 Mnh dn ¼ rh Nf s ð1Þ þ ; 2 dn dn dn dn
R1
ð38Þ
where, Js ðnÞ ¼ A1 b3r1 cosh b1 n þ A4 b3r2 ½b2 ðb22 3b23 Þ sinh b2 n sin b3 n þ b3 ðb23 3b22 Þ cosh b2 n cos b3 n þ A5 b3r2 ½b2 ðb22 3b23 Þ cosh b2 n cos b3 n b3 ðb23 3b22 Þ sinh b2 n sin b3 n
ð39Þ
Jr ðnÞ ¼ B2 b4r1 cosh b1 n þ B3 b4r2 ½ðb42 6b22 b23 þ b23 Þ sinh b2 n sin b3 n 4b2 b3 ðb22 b23 Þ cosh b2 n cos b3 n þ B6 b4r2 ½ðb42 6b22 b23 þ b23 Þ cosh b2 n cos b3 n þ 4b2 b3 ðb22 b23 Þ sinh b2 n sin b3 n
ð40Þ
Because the actuated axial force and the bending moment are the same in the large middle area of the joint, they have been defined as the equivalent axial force and bending moment, which are Nmax ¼ Nnh ð0Þ;
Mmax ¼ Mnh ð0Þ
Eq. (41) defines the non-dimensional equivalence of the actuated forces.
ð41Þ
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Fig. 4. A cantilever beam with the partially bonded PZT.
4.3. The actuated displacements of the host beam In light of the theory of material mechanics (Timoshenko and Gere, 1972), the axial displacement is small as compared with the transverse deflection and thus only the transverse deflection of the beam needs to be solved normally. If necessary, the axial displacement can be simply solved by the constitutive equation as the internal axial force has been found. For the cantilever beam model shown in Fig. 4, which was discussed by Wang and Rogers (1991) without taking into account the adhesive layer, we can obtain 9 a > wnh ¼ 0; 6 n 6 1 > > L > > > > = dwr ð1Þ dws ð1Þ wnh ¼ wr ðnÞ þ ws ðnÞ þ þ ðn þ 1Þ ½wr ð1Þ þ ws ð1Þ; 1 6 n 6 1 ð42Þ dn dn > > > > > > dwr ð1Þ dws ð1Þ > ; þ wnh ¼ 2 n; 1 6 n 6 Lb dn dn where, 12 n2 whr ðnÞ ¼ Mf r ð1Þ Jr ðnÞ ; Enh rh3 2 At n ¼ 1, we have dwr ð1Þ dws ð1Þ wnh ð1Þ ¼ 2 þ dn dn
whs ðnÞ ¼
6 Enh rh2
n2 Js ðnÞ Nf s ð1Þ 2
ð43Þ
ð44Þ
5. PZT used as a sensor When the host beam undergoes mechanical deformation, the bonded PZT patch deforms and the produced electrical charge can be used as a sensing signal. To use the PZT patch as a sensor, we need to derive sensing equations for the smart beam under the action of the applied axial force (F0 ), the transverse force (P0 ) and the bending moment (M0 ) as shown in Fig. 4.
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5.1. Shear stress and peel stress for the applied forces in the host beam Under the action of the applied axial force or the bending moment, the expressions of the shear and peel stresses are the same in format as those under the action of the applied voltage except for the definitions of the following constants. (a) When an axial force (F0 ) is applied to the host beam, we have F0 and k ¼ I; II Enh rh Ga L (b) When a bending moment (M0 ) is applied to the host beam, we find Hnk ¼ ra en ;
Hmk ¼ Hqk ¼ 0
Hnk ¼ ra em ;
Hmk ¼
2rav em ; rh
where; en ¼
Hqk ¼ 0
where; em ¼
6M0 and k ¼ I; II Enh rh2 Ga L2
(c) When the host beam is subjected to a transverse shear force (P0 ) at the free end, we obtain 9 1 1 > > ðHnII þ HnI Þ ¼ ra aL eq ; ðHnII HnI Þ ¼ ra eq > > 2 2 > > > 1 2rav 1 2rav = ðHmII þ HmI Þ ¼ ðHmII HmI Þ ¼ aL e q ; eq 2 2 rh rh > > > > > > 1 2rav 1 > ; ðHqII þ HqI Þ ¼ ðHqII HqI Þ ¼ 0 eq ; 2 2 rh
ð45Þ
ð46Þ
ð47Þ
where, b aL ¼ ; L
eq ¼
6P0 Enh rh2 Ga L
ð48Þ
In this case, distributions of the shear stress and the peel stress are different from those of the applied axial force or bending moment. In additional to the anti-symmetric shear stress and symmetric peel stress components, there exist the symmetric shear stress and anti-symmetric peel stress components and are given by snsym ¼ A2 cosh b1 n þ A3 sinh b2 n sin b3 n þ A6 cosh b2 n cos b3 n þ A7 ð49Þ rnant ¼ B1 sinh b1 n þ B4 cosh b2 n sin b3 n þ B5 sinh b2 n cos b3 n However, it is worth noting that the sensing electric charge is produced only by the anti-symmetric shear stress and symmetric peel stress components. 5.2. Sensing equation Lee (1992) derived the following sensor equation: Z Z ou0 ov0 ou0 ov0 þ e36 qk ¼ e31 þ e32 þ þ eT33 E3 dx dy ox oy oy ox S ð12Þ Z Z o2 w o2 w o2 w e31 2 þ e32 2 þ 2e36 dx dy z0k ox oy oxoy S ð12Þ
ð50Þ
where, qk is the electric charge on layer k of the laminates, and z0k ¼ ðzk þ zk1 Þ=2 is the half thickness of that layer.
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Substituting the non-dimensional displacements of the beam into Eq. (50), we find Z 1 dun1 zk þ zk1 d2 wn1 qn ¼ e31 dn dn 2 dn2 1 where, qn ¼ q=L; q is the electric charge on the surface of PZT. By employing the constitutive relations of the PZT, we have Z 1 e31 qn ¼ Nn1 dn En1 r1 1
ð51Þ
ð52Þ
Substituting the exact solutions of the shear stress produced by the applied forces, we can obtain the following sensing equation: 2e31 dJs ð1Þ qn ¼ Nf s ð1Þ ð53Þ dn En1 r1 6. The host beam with two PZTs bonded symmetrically on both sides Crawley and de Luis (1987), and Im and Atluri (1989) investigated the active behaviors of the smart beam with the bonded PZT symmetrically. It is assumed that the poling direction of the PZT patch on the top surface is aligned with that of the PZT patch on the lower surface. When a pair of PZT patches is
Fig. 5. FBD for each element of the host beam with two PZT patches: (a) a smart beam with two symmetrically bonded PZT patches, (b) the PZT1 infinitesimal element, (c) the infinitesimal element of adhesive layer 1, (d) the infinitesimal element of the host beam, (e) the inifinitesimal element of adhesive layer 2 and (f) the PZT2 infinitesimal element.
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bonded to the top and bottom surfaces of the host beam symmetrically, as shown in Fig. 5, we can obtain the following non-dimensional equilibrium and constitutive equations. (a) The equilibrium equations are dNn1 þ sn1 ¼ 0; dn
dQn1 þ rn1 ¼ 0; dn
dNnh þ sn2 sn1 ¼ 0; dn dNn2 sn2 ¼ 0; dn
dMn1 1 þ r1 sn1 Qn1 ¼ 0 2 dn
dQnh þ rn2 rn1 ¼ 0; dn
9 > > > > > > > =
dMnh 1 þ rh ðsn2 þ sn1 Þ Qnh ¼ 0 > 2 dn > > > > > dMn2 1 > ; þ r1 sn2 Qn2 ¼ 0 2 dn
dQn2 rn2 ¼ 0; dn
ð54Þ
(b) By referring to Eqs. (6) and (7) and Fig. 5, the stress–displacement relations of the adhesive layers are 9 Rta dwnh dwn1 > > sn1 ¼ ra ðunh un1 Þ þ Rht þ ; rn1 ¼ rav ðwnh wn1 Þ > = 2 dn dn ð55Þ > Rta dwnh dwn2 > > Rht þ sn2 ¼ ra ðun2 unh Þ þ ; rn2 ¼ rav ðwn2 wnh Þ ; 2 dn dn (c) The constitutive equations of the host beam and PZT are 9 dun1 e31 V1 dunh dun2 e31 V2 > > Nn1 ¼ En1 r1 þ ; Nnh ¼ Enh rh ; Nn2 ¼ En1 r1 þ ; > = dn Ga L dn dx Ga L 1 d2 wn1 1 d2 wnh 1 d2 wn2 > > > Mn1 ¼ En1 r13 ; Mnh ¼ Enh rh3 ; Mn2 ¼ En1 r13 2 2 2 ; 12 12 12 dn dn dn
ð56Þ
The symbol definitions are the same as those of the host beam with PZT on its top surface, and i ¼ 1; 2; h, in which subscripts 1 and 2 refer to the top and lower PZT patches and subscript h refers to the host beam. Eqs. (54)–(56) can be transferred into two sets of the independent equations for extension and bending that can be solved separately. Case 1. [Host beam deformed in extension] When the host beam undergoes extension only, we have the following set of equations, in which, the following transforming parameters are used: 2Ns ¼ Nn1 þ Nn2 ; 2Qs ¼ Qn1 Qn2 ; 2us ¼ un1 þ un2 ; 2ws ¼ wn1 wn2 2ss ¼ sn1 sn2 ;
2rs ¼ rn1 þ rn2 ;
9 2Ms ¼ Mn1 Mn2 > =
2Vs ¼ V1 þ V2
> ;
ð57Þ
The equilibrium equations are dNs dQs þ ss ¼ 0; þ rs ¼ 0; dn dn dNnh 2ss ¼ 0 dn
9 dMs 1 þ r1 ss Qs ¼ 0 > = 2 dn > ;
ð58Þ
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The relations between the stresses/stress resultants and displacement are given by 9 dus e31 Vs 1 d2 ws dunh > > Ns ¼ En1 r1 þ ; Ms ¼ En1 r13 ; N ¼ E r = nh nh h 12 dn Ga L dn dn2 ð59Þ > Rta dws > ; ss ¼ ra ðunh us Þ þ ; rs ¼ rav ws 2 dn When both PZTs are used as actuators, extensional deformation can be achieved by applying a voltage (Vs ) of the same magnitude to both PZT patches so that the electric fields in both PZTs are in the same direction. Case 2. [Host beam deformed in bending] When the host beam undergoes the pure bending deformation, the following transforming parameters are used: 9 2Na ¼ Nn1 Nn2 ; 2Qa ¼ Qn1 þ Qn2 ; 2Ma ¼ Mn1 þ Mn2 = ð60Þ 2ua ¼ un1 un2 ; 2wa ¼ wn1 þ wn2 ; 2sa ¼ sn1 þ sn2 ; 2ra ¼ rn1 rn2 ; 2Va ¼ V1 V2 The equilibrium equations are 9 dNa dQa dMa 1 = þ sa ¼ 0; þ ra ¼ 0; þ r1 sa Qa ¼ 0 > 2 dn dn dn dQnh dMnh > ; 2ra ¼ 0; þ rh sa Qnh ¼ 0 dn dn
ð61Þ
and the stress/stress resultant–displacement relationships are given as
9 2 dua e31 Va 1 d2 wa 1 3 d wnh > > Na ¼ En1 r1 E þ ; Ma ¼ En1 r13 ; M ¼ r = nh nh h 12 12 dn G dn2 dn2 aL > Rta dwnh dwa > ; Rht þ s a ¼ r a ua þ ; ra ¼ rav ðwnh wa Þ 2 dn dn
ð62Þ
When both PZTs are used as actuators, bending deformation can be achieved by applying a voltage (Va ) of the same magnitude but opposite directions to both PZT patches so that the electric fields in both PZTs have the same magnitude but opposite directions. The equations for Case 1 can be transformed into the coupled ODE as shown in Eq. (8), in which 9 Rta 4 2 6Rta > > kn1 ¼ ks1 ¼ 2 þ ; kn2 ¼ ks2 ¼ = r1 En1 Enh Rht En1 r13 ð63Þ rav 12rav Rta > > kn3 ¼ ks3 ¼ ks2 ; kn4 ¼ ks4 ¼ ; ra En1 ra r14 Similarly, the coupled ODE as given in Eq. (8) can also be derived for Case 2, in which 9 Rta 4 6 6Rta 1 2 > > kn1 ¼ ka1 ¼ 2 þ ; kn2 ¼ ka2 ¼ 3 > En1 Enh R2ht = r1 En1 Enh Rht r1 ð64Þ > rav 12rav Rta 1 2 > > þ kn3 ¼ ka3 ¼ ka2 ; kn4 ¼ ka4 ¼ ; En1 Enh R3ht ra ra r14 Solutions to Cases 1 and 2 can also be expressed as Eqs. (15) and p (16). ffiffiffi When Enh R2ht En1 ¼ 0, for example, it is equivalent to that h ¼ 2t1 for the same elastic moduli of the host beam and PZT. In this case, ka2 ¼ ka3 ¼ 0 and the coupled ODEs become decoupled for Case 2: d 3 sa d4 ra 2 dsa ; ¼ b ¼ b4ua2 ra ua1 dn dn3 dn4 pffiffiffiffiffiffi pffiffiffiffiffiffi where, bua1 ¼ ka1 , bua2 ¼ 4 ka4 .
ð65Þ
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Solution to Eq. (65) is similar to that of Eq. (10) with the integration constants given in Appendix A. The polynomials for determining the eigenvalues can be obtained by replacing kni (i ¼ 1–4) with kri (r ¼ s, a; i ¼ 1–4) and solutions to the shear stress and peel stress are also similar to those of the host beam with the bonded PZT on the top surface. The boundary conditions can be expressed as d2 wsk 12Msk d3 wsk 6ssk 12Qsk ¼ ; ¼ ; 2 3 3 2 E r E r En1 r13 dn dn n1 1 n1 1 du N d2 wak 12Mak d3 wak 6ssk 12Qsk n ¼ 1 : ak ¼ ak þ eae ; ¼ ; ¼ ; 2 3 3 2 dn En1 r1 En1 r1 En1 r13 En1 r1 dn dn dunhk Nnhk d2 wnhk 12Mnhk d3 wnhk 12sak 12Qnhk ¼ ; ¼ ; ¼ dn Enh rh Enh rh2 Enh rh3 Enh rh3 dn2 dn3 dusk Nsk ¼ þ ese ; dn En1 r1
e31 Vs ese ¼ E1 t 1
9 > > > > > > > > =
e31 Va E1 t 1 > > > > > > > ; ðk ¼ I; IIÞ >
eae ¼
ð66Þ
in which, ese and eae are the electrically actuated strains. The boundary conditions expressed in terms of shear and peel stresses for Case 1 are: n ¼ 1 :
dss ¼ Hnsk ; dn
d2 rs ¼ Hmsk ; dn2
d3 rs þ ks3 ss ¼ Hqsk dn3
ðk ¼ I; IIÞ
ð67Þ
where, 9 Nsk Nnhk 6Rta Msk > > Hnsk ¼ ra þ ese þ > En1 r1 Enh rh En1 r13 = ðk ¼ I; IIÞ > 12rav Msk 12rav Qsk > > Hmsk ¼ ; Hqsk ¼ ; En1 r13 En1 r13
ð68Þ
The boundary conditions expressed in terms of shear and peel stresses for Case 2 are: n ¼ 1 :
dsa ¼ Hnak ; dn
d2 ra ¼ Hmak ; dn2
d3 ra þ ka3 ss ¼ Hqak dn3
ðk ¼ I; IIÞ
ð69Þ
where,
9 Nak Mnhk Mak > > Hnak ¼ ra þ eae þ 6Rta Rht þ > > = En1 r1 Enh rh3 En1 r13 ðk ¼ I; IIÞ > Mak Mnhk Qak Qnhk > > > Hmak ¼ 12rav ; Hqak ¼ 12rav ; En1 r13 Enh rh3 En1 r13 Enh rh3
ð70Þ
We now obtain the same forms of the shear stress, the peel stress and the boundary conditions in Cases 1 and 2 for the smart beam with the bonded PZTs symmetrically, and they are also the same as those obtained for the smart beam with the PZT patch on the top surface. Therefore, the developed solutions for the case with the PZT on the top surface can be readily employed. Having solved the ss ; rs and sa ; ra , we can obtain the shear and peel stresses in the adhesive layers: sn1 ¼ ss þ sa ;
sn2 ¼ sa ss
and rn1 ¼ rs þ ra ;
rn2 ¼ rs ra
ð71Þ
Using the similar procedures to those used in the host beam with one PZT on the top surface, the actuated internal forces, displacements and the sensing electric charges can be derived. Because the forms of the equilibrium equations, constitutive relations and boundary conditions are the same as those in the case with the PZT on the top surface, the related formulations can also be used directly.
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When the PZTs are used as actuators and the same voltages are applied to both PZTs with the same poling direction, the host beam will be deformed in extension and the pure bending occurs for the same value but opposite voltages. The phenomena are similar to those predicted by Crawley and de Luis (1987), but the values are different for flexible structures, which will be discussed in Part II.
7. Conclusion This paper takes into account the peel stress in the mathematical formulations for smart beams with PZT actuators and sensors. Exact solutions for both shear and peel stresses are developed in general and explicit form. The exact solutions are then used to develop analytical solutions for two cases: (a) a cantilever beam with one PZT patch bonded to one surface of the host beam; and (b) a cantilever beam with two PZT patches symmetrically-bonded both surfaces of the host beam. Exact formulations are obtained for the actuated force, bending moment and displacement as well as sensing equation. A mathematical formulation is also obtained for the conditions when the coupled peel and shear equations can be decoupled.
Acknowledgements The authors are grateful to the support of the Australian Research Council through a Large Grant Scheme (grant no. A10009074). Appendix A. Solutions to the integration constants Solutions to Eqs. (26) and (28) are as follows: 1 HnII þ HnI HmII þ HmI HqII HqI A1 ¼ bn bm þ bq D 2 2 2 1 A4 ¼ ½ðb cosh b2 cos b3 bn3 sinh b2 sin b3 ÞðHnII þ HnI Þ Dbq n2 ðbm2 cosh b2 cos b3 bm3 sinh b2 sin b3 ÞðHmII þ HmI Þ þ ðb12 cosh b2 cos b3 b13 sinh b2 sin b3 Þbq cosh b1 ðHqII HqI Þ 1 ½ ðbn2 sinh b2 sin b3 þ bn3 cosh b2 cos b3 ÞðHnII þ HnI Þ A5 ¼ Dbq
þðbm2 sinh b2 sin b3 þ bm3 cosh b2 cos b3 ÞðHmII þ HmI Þ ðb12 sinh b2 sin b3 þ b13 cosh b2 cos b3 Þbq cosh b1 ðHqII HqI Þ
9 > > > > > > > > > > > > > > > > > > =
> > > > > > > > > > > > > > > > > > ;
ðA:1Þ
where,
9 D ¼ bL1 bq sinh b1 þ bD cosh b1 > > > bn ¼ ðbK2 bL3 þ bK3 bL2 Þ sinh 2b2 þ ðbK2 bL2 bK3 bL3 Þ sin 2b3 > > > > > > bm ¼ ðb2 bL3 þ b3 bL2 Þ sinh 2b2 ðb2 bL2 b3 bL3 Þ sin 2b3 > > = bq ¼ ðb3 bK2 b2 bK3 Þðcosh 2b2 þ cos 2b3 Þ bD ¼ ðb12 bL3 þ b13 bL2 Þ sinh 2b2 þ ðb12 bL2 b13 bL3 Þ sin 2b3 > > > > > bn2 ¼ DbK2 bn b12 cosh b1 ; bn3 ¼ DbK3 bn b13 cosh b1 > > > > bm2 ¼ Db2 þ bm b12 cosh b1 ; bm3 ¼ Db3 þ bm b13 cosh b1 > > ; b12 ¼ bK1 b2 b1 bK2 ; b13 ¼ bK1 b3 b1 bK3
ðA:2Þ
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9 > > > > > > > > > > > > > > > > > > > =
K1 HnII þ HnI HmII þ HmI HqII HqI bn bm þ bq D 2 2 2 1 B3 ¼ f½ðK2 bn2 K3 bn3 Þ cosh b2 cos b3 ðK2 bn3 þ K3 bn2 Þ sinh b2 sin b3 ðHnII þ HnI Þ Dbq
B2 ¼
½ðK2 bm2 K3 bm3 Þ cosh b2 cos b3 ðK2 bm3 þ K3 bm2 Þ sinh b2 sin b3 ðHmII þ HmI Þ þ ½ðK2 b12 K3 b13 Þ cosh b2 cos b3 ðK2 b13 þ K3 b12 Þ sinh b2 sin b3 bq ðHqII HqI Þ
> > > > > > 1 > B6 ¼ f ½ðK3 bn2 þ K2 bn3 Þ cosh b2 cos b3 ðK3 bn3 K2 bn2 Þ sinh b2 sin b3 ðHnII þ HnI Þ > > > Dbq > > > > > > þ½ðK3 bm2 þ K2 bm3 Þ cosh b2 cos b3 ðK3 bm3 K2 bm2 Þ sinh b2 sin b3 ðHmII þ HmI Þ > > > ; ½ðK3 b12 þ K2 b13 Þ cosh b2 cos b3 ðK3 b13 K2 b12 Þ sinh b2 sin b3 bq ðHqII HqI Þ 1 HnII HnI HmII HmI HqII þ HqI kn3 ðNn1II Nn1I Þ A2 ¼ bm0 þ bq0 bn0 D0 2 2 2 2 1 A3 ¼ fðbn02 sinh b2 cos b3 bn03 cosh b2 sin b3 ÞðHnII HnI Þ D0 bq0 ðbm02 sinh b2 cos b3 bm03 cosh b2 sin b3 ÞðHmII HmI Þ
9 > > > > > > > > > > > > > > > > > > > =
þ ðb12 sinh b2 cos b3 b13 cosh b2 sin b3 Þbq0 sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ > > > > > > 1 > > A6 ¼ f½ðbn02 cosh b2 sin b3 þ bn03 sinh b2 cos b3 ÞðHnII HnI Þ > > D0 bq0 > > > > > > þðbm02 cosh b2 sin b3 þ bm03 sinh b2 cos b3 ÞðHmII HmI Þ > > > ; ðb12 cosh b2 sin b3 þ b13 sinh b2 cos b3 Þbq0 sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ A7 ¼
NnI1I Nn1I HnII HnI HmII HmI þ ðbr1 bn0 sinh b1 þ brn Þ þ ðbr1 bm0 sinh b1 brm Þ 2 2D0 2D0 HqII þ HqI kn3 ðNn1II Nn1I Þ ðbr1 brq Þbq0 sinh b1 2D0 2D0
ðA:3Þ
ðA:4Þ
ðA:5Þ
where, 9 > > > > > bn0 ¼ ½bK2 bL3 þ bK3 ðbl21 sin2 b3 þ bl22 cos2 b3 Þ sinh 2b2 > > > > 2 2 > þ½bK3 bL3 þ bK2 ðbl21 sinh b2 bl22 cosh b2 Þ sin 2b3 > > > > 2 > 2 > bm0 ¼ ½b2 bL3 þ b3 ðbl21 sin b3 þ bl22 cos b3 Þ sinh 2b2 > > > 2 2 > þ½b2 ðbl22 cosh b2 bl21 sinh b2 Þ b3 bL3 sin 2b3 > > > > = b ¼ ðb b b b Þðcosh 2b cos 2b Þ D0 ¼ bL11 bq0 cosh b1 þ bD0 sinh b1
q0
3 K2
2 K3
2
3
> > > > > 2 2 > þ½b13 bL3 þ b12 ðbl21 sinh b2 bl22 cosh b2 Þ sin 2b3 > > > > > bn02 ¼ D0 bK2 bn0 b12 sinh b1 ; bn3 ¼ D0 bK3 bn0 b13 sinh b1 > > > > > bm02 ¼ D0 b2 þ bm0 b12 sinh b1 ; bm03 ¼ D0 b3 þ bm0 b13 sinh b1 > > > > > > b3 bn02 þ b2 bn03 > > brn ¼ 2 ; 2 ðb2 þ b3 Þðb3 bK2 b2 bK3 Þ 2
bD0 ¼ ½b12 bL3 þ b13 ðbl21 sin b3 þ bl22 cos2 b3 Þ sinh 2b2
ðA:6Þ
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K1 HnII HnI HmII HmI HqII þ HqI kn3 ðNn1II Nn1I Þ bn0 bm0 þ bq0 2 D 2 2 2 1 B4 ¼ f½ðK2 bn02 K3 bn03 Þ sinh2 cos b3 ðK2 bn03 þ K3 bn02 Þ cosh b2 sin b3 ðHnII HnI Þ D0 bq0
B1 ¼
½ðK2 bm02 K3 bm03 Þ sinh b2 cos b3 ðK2 bm03 þ K3 bm02 Þ cosh b2 sin b3 ðHmII HmI Þ
9 > > > > > > > > > > > > > > > > > > > > > > > =
þ½ðK2 b12 K3 b13 Þ sinh b2 cos b3 ðK2 b13 þ K3 b12 Þ cosh b2 sin b3 bq0 sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ > > > > > 1 > > B5 ¼ ½ðK b þ K b Þ sinh b cos b ðK b K b Þ cosh b sin b ðH H Þ f 3 n02 2 n03 3 n03 2 n02 nII nI > 2 3 2 3 > > D0 bq0 > > > > > þ½ðK3 bm02 þ K2 bm03 Þ sinh b2 cos b3 ðK3 bmo3 K2 bmo2 Þ cosh b2 sin b3 ðHmII HmI Þ > > > > > ½ðK3 b12 þ K2 b13 Þ sinh b2 cos b3 ðK3 b13 K2 b12 Þ cosh b2 sin b3 > > > ; bq sinh b1 ½ðHqII þ HqI Þkn3 ðNn1II Nn1I Þ ðA:7Þ If the substructures have the same material properties and the geometric sizes, solutions to the integration constants are 9 HnII þ HnI > A1 ¼ > > > 2bsn1 cosh bsn1 > > > ðsinh be cos be þ cosh be sin be Þbe ðHmII þ HmI Þ sinh be sin be ðHqII HqI Þ = B1 ¼ ðA:8Þ 2b3e ðsinh 2be þ sin 2be Þ > > > > ðsinh be cos be cosh be sin be Þbe ðHmII þ HmI Þ cosh be cos be ðHqII HqI Þ > > > ; B4 ¼ 2b3e ðsinh 2be þ sin 2be Þ A2 ¼
HnII HnI ; 2bsn1 sinh bsn1
A3 ¼
Nn1II Nn1I HnII HnI 2 2b2sn1
9 ðsinh be sin be þ cosh be cos be Þbe ðHmII HmI Þ cosh be sin be ðHqII þ HqI Þ > > > > B2 ¼ = 2b3e ðsinh 2be sin 2be Þ ðcosh be cos be sinh be sin be Þbe ðHmII HmI Þ sinh be cos be ðHqII þ HqI Þ > > > B3 ¼ > ; 2b3e ðsinh 2be sin 2be Þ
ðA:9Þ
ðA:10Þ
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