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An analytical model of constrained piezoelectric thin film sensors
Sensors and Actuators A 116 (2004) 424–437
An analytical model of constrained piezoelectric thin film sensors R. Ali, D. Roy Mahapatra, S. Gopalakrishnan∗ Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India Received 6 October 2003; received in revised form 28 April 2004; accepted 2 May 2004 Available online 8 July 2004
Abstract A new analytical model based on 2D electro-mechanical continuum model is developed to analyze the performance of constrained piezoelectric thin film sensors, surface-bonded or embedded in composite material system. Several non-classical effects, such as viscoelastic property of the bonding layer, AC dielectric loss, frequency-dependent effect of the resistivity of the film materials, and most importantly, the effect of constrained boundary on the capacitive performance are considered. The model is validated by comparing the results from electrostatics as well as a simplified model based on direct stress transfer and strain continuity at the film-substrate interface. Analytical solutions for specific cases of boundary constraints on the film, such as in-plane tensile stress, transverse normal stress and horizontal shear stress are reported. Numerical studies on the effect of these stresses on PVDF and PZT films with carbon-epoxy host structure are carried out. Effect of process-induced residual stress is also studied. The results show significant complexity, which is otherwise intractable using existing simplified approaches. © 2004 Elsevier B.V. All rights reserved. Keywords: Constrained piezoelectric thin film; MEMS; Capacitance; Residual stress; Voltage
1. Introduction This paper discusses a new analytical model of constrained piezoelectric thin film sensors, surface-bonded or embedded in a composite structure. The approach is one step improvement over the already prevailing analysis and design methodology for conventional micro-electro-mechanical transducers [1]. The state of boundary constraints on piezoelectric film sensor is important to account in many applications. For example, in structural health monitoring (SHM) applications, the identification of hidden structural defects and any damage process taking place at sub-surface and/or in deep inter-laminar regions in composites using film sensor are of great importance [2–4]. Conventional strain gauges and MEMS sensor arrays are being thought of performing the damage detection task. Another example is the electro-active membrane type adaptive structures for shape and vibration control applications, where the understanding of the effect of various material and geometric parameters on the electro-mechanical coupling is essential. One of the major concerns for MEMS is its relatively poor reliability caused by the delamination, brittle fracture and
∗ Corresponding author. Tel.: +91 80 22933019; fax: +91 80 23600134. E-mail address: [email protected] (S. Gopalakrishnan).
0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.05.028
fatigue degradation of multi-layer thin film structures [5,6]. However, the task of sensing the deformation pattern (e.g. damage detection) can be performed best if the film of large planer dimension with segmented electrode arrays can be modeled, designed and fabricated. While validating such design concept, it is essential to incorporate additional effects, such as, the electro-mechanical coupling in the sensor structure, the dissipation and the electrical resistivity of the sensor material, AC frequency, viscoelastic loss in the adhesive matrix, as against the conventional design [1] where all these effects are not taken into consideration. A major problem with adhesively bonded wafer, or more significantly with deposited thin film, is the control of the state of mechanical stress. Various sources of stress may be discerned when a thin film is deposited or formed by reaction on a substrate [7]. This stress, namely the residual stress, is induced inevitably in thin film due to structural change and thermal misfit between the film and the substrate and the change from high to low temperature during the process [8]. The residual stress significantly affects the service-life of the film, and hence of the whole device itself [8,9]. The residual tensile stress may cause surface crack or tunneling crack in the film [10,11]. The residual compressive stress may cause the film to delaminate from the substrate. The residual stress mainly comprises of Epitaxial, Intrinsic, Transformation and Thermal stresses [7,8]. The strains as-
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
sociated with each of these stresses can be obtained as discussed in the works reported in [12]. With the introduction of simplified interface compliance, both the thermal, as well as the interfacial stresses between the film and the substrate can be estimated [13,14]. In order to design and fabricate better transducer devices with higher reliability, the effect of thickness and length of the piezoelectric film, substrate properties and misfit strain on the distributions of transverse and in-plane stresses must be precisely known. Since the lattice of the thin film and substrate shall be matched with each other during film growth, a misfit strain will be introduced at the interface [15]. The model proposed in this paper will be capable of handling such pre-estimated residual stress and misfit strains while analyzing the capacitive performance of the device. Most of the analytical and finite element analysis technique for coupled electro-mechanical simulations reported in literature are based on iteration over mechanical and electrical equivalent of stress field to achieve optimum capacitance and voltage (e.g. see [16]). In recent time, there has been increasing effort in developing analytical technique to study the constrained effect on thin films [15,17]. Similar problems in distributed transducer application is also expected to play important role. Hence, a systematic analytical model of the coupled electro-mechanical field appears advantageous. In the present paper, we develop a two dimensional electro-mechanical continuum field model to analyze the effects of boundary constraints on the transduction performance of thin as well as thick piezoelectric films. The analytical model proposed in this paper accommodates plane stress or plane strain constitutive properties of piezoelectric layer across the layer thickness, AC frequency dependent dielectric properties, viscoelastic bonding layer and homogeneous substrate material properties. Experimentally observed residual stress field can also be incorporated towards achieving a better design methodology. Closed-form analytical solutions for the capacitance and voltage are derived under various boundary constraints. Numerical results are presented by considering a range of layer thicknesses, lengths and stresses on PVDF and PZT films. To begin with the numerical studies, a database was collected for different sets of materials used for the host structure and the film. Among these sets, results are presented for carbon-epoxy composite as host structural material, and PVDF or PZT as film material. Parametric variation of capacitance and sensor voltage are studied in detail. The results are compared with those used in conventional design. The voltage variation over the length of the film is shown for several cases of direct tension, and surface tractions. The effect of increasing stress level on the capacitance, within the limit of the structural failure, is also presented. The surface-bonded film performance is simulated as a special case of embedded film. Its performance is compared with that of a film based on a very simplistic approach of stress transfer and strain continuity at the surface of the host structure.
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2. Piezoelectric constitutive model A plane stress case in the plane (X, Z) of the film embedded in host structure is considered here. Z indicates the thickness coordinate of the film. The schematic free body diagram of the sensor is shown in Fig. 1. Under plane stress condition, the 3D constitutive model of the bulk piezoelectric materials [18] can be reduced to 2D in (X, Z) plane by imposing the out-of-plane stresses σyy = 0, τxy = 0, τyz = 0 and the electric charge displacements in the plane (X, Z), Dy = 0 and in the plane (Y, Z), Dx = 0. As a result, the 2D constitutive model can be expressed as a εaxx σ c c 0 −e 11 13 31 xx a σa c εzz c 0 −e 13 33 33 zz a = (1) γ a xz , 0 τxz 0 c55 0 ∂Φ Dz e31 e33 0 33 ∂z where the superscript a indicates sensor layer, cij are the plane-stress reduced elastic moduli, eij are the plane-stress reduced electro-mechanical coupling coefficients, 33 is the dielectric constant of the piezoelectric film, Dz is the electric charge displacement causing deformation-induced charge accumulation on the film surfaces parallel to (X, Y ) plane. The associated conjugate electric field is ∂Φ/∂z, where Φ is the electric potential across the electrodes parallel to the (X, Y ) plane. Here the strain εxx comprises of the strain due to elastic deformation εdxx and the residual strain εrxx . Therefore, the total strain along X-axis is given by εaxx = εdxx +εrxx . Note that in Eq. (1), the coupling between the shear strain and the electric field is neglected. Such constitutive model is applicable to transversely polarized piezoelectric film. The governing equations can be expressed as a ∂τ a ∂σxx + xz = 0, ∂x ∂z
(2a)
a ∂τ a ∂σzz + xz = 0, ∂z ∂x
(2b)
∂Dz = 0. ∂z
(2c)
Here, Eq. (2c) is the reduced form of one of the Maxwell’s equation ∇×D = ρ for material medium including the effect of polarization, where ρ is the total charge density. Under the Z
+ σzz
τ+xz tm σxx
ta
X
Sensor layer
tm
τ xz
Bonding layer Substrate
σ zz
Fig. 1. Schematic diagram of the embedded sensor.
σ+xx
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closed-circuit condition, the accumulated charge on the film surfaces, which is due to the deformation, dissipates across the interface electronic circuit. Hence, for simplicity and by assuming negligible residual charge, one has Eq. (2c). Using Eq. (1) in Eqs. (2a)–(2c),
G∗ is the complex shear modulus of the bonding layer by taking into account its viscoelastic property. Assuming the residual strain as constant over the film length, the strains can be expressed in terms of the displacements as
∂εa ∂γ a ∂εa ∂2 Φ c11 xx + c13 zz + c55 xz − e31 = 0, ∂x ∂x ∂z ∂x∂z
(3a)
εaxx =
∂εa ∂γ a ∂εaxx ∂2 Φ + c33 zz + c55 xz − e33 2 = 0, ∂z ∂z ∂x ∂z
(3b)
∂εa ∂εaxx ∂2 Φ + e33 zz + 33 2 = 0. ∂z ∂z ∂z
(3c)
c13 e31
Let ua and wa be the displacements in the fim along the X-axis and Z-axis respectively. Since the film thickness is much smaller compared to its length, we can assume the inplane and bending deformations to be dominant compared to the transverse shear deformation. Assuming first order shear deformation across the film thickness, ua can be given by (5)
where u0 is the film mid-plane (z = 0) displacement of the sensor. φ is the rotation of the vertical cross-section about Y -axis. Eq. (5) can further be approximated in terms of the applied displacements and traction on the upper and lower surfaces of the sensor embedded in host substrate through the bonding layer (see Fig. 1), which yields u+ + u− tm + − − (τ + τxz ), 2 2G∗ xz u+ − u− tm + − φ= − ∗ (τxz − τxz ), ta G ta
u0 =
(6)
where u+ and u− are respectively the horizontal displacements at the upper and the lower surfaces (z = ±(tm +ta /2)) + and τ − are respectively the horizontal of the sensor. τxz xz shear stresses at the upper and the lower surfaces (z = ±(tm + ta /2)) of the sensor. ta and tm are respectively the thicknesses of the film and the bonding layer. Here, the bonding layer is assumed to be an equivalent single layer including the electrodes, insulation layer, adhesive layer etc.
∂wa , ∂z
a =φ+ γxz
∂ 2 wa ∂ 2 wa ∂φ + k + (c55 + k1 ) = 0, 2 2 2 ∂x ∂x ∂z
where e31 e33 k1 = c13 + , 33
The steps involved in obtaining the proposed analytical solution for the constrained piezoelectric film shown in Fig. 1 are discussed below. Eliminating Φ from Eq. (3b) using Eq. (3c), one can write a ∂γxz e31 e33 a ∂ c55 εxx + c13 + ∂x ∂z 33 2 e33 + c33 + εa = 0 (4) 33 zz
εazz =
∂wa . ∂x (7)
Substituting the above, Eq. (4) can be simplified as c55
3. Analytical solution
ua = u0 + zφ,
∂u0 ∂φ +z + εrxx , ∂x ∂x
e2 k2 = c33 + 33 33
(8) .
(9)
Eq. (8) is the governing equation for transverse bending and shear in the film. Observing this equation, wa can be expressed as a function of φ, which is given by c55 + k1 wa = − φ dx + C1 x + C2 + z(C3 + C4 x), c55 (10) where C1 , C2 , C3 , C4 are all constants of integration. Subsequently, ∂wa = −αφ + C1 + C4 z, ∂x where k1 α= 1+ . c55
∂wa = C3 + C4 x, ∂z
(11)
(12)
The objective is now to eliminate the constants Cj with the help of appropriate boundary conditions at the film edges, the upper and the lower surfaces. Since we have assumed that the transduction takes place due to the deformation induced electric potential Φ across the parallel segmented electrodes at the upper and lower surfaces of the film, there are no electrodes or material discontinuities at the two vertical cross-sections of the film at x = ±L/2, where L is the film length. On these two vertical surfaces, the normal traction is given by 0 ∂u ∂φ a σxx +z + εrxx + c13 (C3 + C4 x) . = c11 (13) ∂x ∂x + Considering the cross-sectional average normal stresses σxx − and σxx at the mid-points of the vertical surfaces of the film respectively at x = ±L/2, z = 0 (see Fig. 1), we get two boundary conditions 0 L ∂u + c11 + εrxx = σxx + c13 C3 + C4 , (14a) ∂x 2 x=L/2 0 L ∂u r − c11 + c13 C3 − C4 . (14b) + εxx = σxx ∂x 2 x=−L/2
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
From the above two equations, one can evaluate C3 and C4 as 1 ∂u0 + − σxx + σxx − c11 C3 = 2c13 ∂x x=L/2 ∂u0 r + , (15a) + 2εxx ∂x x=−L/2 1 ∂u0 + − σ − σxx − c11 C4 = Lc13 xx ∂x x=L/2 ∂u0 − . ∂x x=−L/2
(15b)
To evaluate the rest of the two constants C1 and C2 , we assume wa = 0 at the lower end corners i.e. at x = ±L/2, z = −ta /2. This is a typical restraint to model the four-point bending of a film and also can be used to model (1) pure horizontal shear, (2) thin film on a stiff substrate subjected to surface horizontal traction and (3) thickness-wise symmetric embedded film/wafer and host structure subjected to pure tension/compression. Substituting these two displacement boundary conditions in Eq. (10) and simplifying, we get C4 ta α C1 = , − φ dx + φ dx 2 L x=L/2 x=−L/2 (16a) α C3 ta + C2 = 2 2
φ dx
x=L/2
+
φ dx
x=−L/2
. (16b)
Now, the displacement field can be described explicitly with the help of Eqs. (6) and (10). Further, they can be used to explicitly describe the strain field in Eq. (7). Simplifying the electric displacement Dz in Eq. (1) with the help of the explicit form of the strain field and evaluating at z = ta /2, we get 0 ∂u ta ∂φ r D+ = e + + ε 31 z xx + e33 (C3 + C4 x) ∂x 2 ∂x ∂Φ + 33 . (17) ∂z z=ta /2 In order to eliminate the electric potential Φ from Eq. (17), we use one additional boundary condition, that is the normal + (x) applied on the upper surface (z = t /2) of traction σzz a the film. With the help of Eq. (1) one can write 0 ∂Φ 1 ∂u ta ∂φ + r = −σzz (x) + c13 + + εxx ∂z z=ta /2 e33 ∂x 2 ∂x c33 + (C3 + C4 x). (18) e33
Substituting back in Eq. (17), we get 0 ∂u c13 33 ta ∂φ r = e + + + ε D+ 31 z xx e33 ∂x 2 ∂x + (x) 33 σzz c33 33 + e33 + (C3 + C4 x) − . e33 e33
427
(19)
The next objective is to obtain the explicit form of the potential difference between the upper and lower surface of the film, so that the capacitance of the film under various types of boundary constraints can be estimated. In order to do this, we shall reconsider the governing Eqs. (3a)–(3c). Out of these three equations, since Eqs. (3b) and (3c) have already been used, we shall now use Eq. (3a) to derive the potential difference. Using the expressions for strains given by Eq. (7) and substituting in Eq. (3a), we get 2 0 ∂ u ∂2 φ ∂2 Φ c11 + C + z (c + c ) − e = 0. 4 13 31 55 ∂x∂z ∂x2 ∂x2 (20) Integrating Eq. (20) once with respect to x, 0 ∂u ∂φ ∂Φ c11 +z + (c13 + c55 )C4 x − e31 = D1 , ∂x ∂x ∂z (21) where D1 is the constant of integration. D1 can be eliminated by evaluating Eq. (21) at x = L/2, z = ta /2 as 0 ∂u c13 e31 ta ∂φ D1 = c11 − + e33 ∂x 2 ∂x x=L/2 C4 L c33 e31 + c13 + c55 − 2 e33 e31 + c13 e31 r (σ (x)|x=L/2 − c33 C3 ) − + ε . (22) e33 zz e33 xx Integrating Eq. (21) once with respect to z, ∂u0 z2 ∂φ c11 z + + (c13 + c55 )C4 xz − e31 Φ ∂x 2 ∂x = D1 z + D2 ,
(23)
where D2 is another constant of integration. Let the electric potential at the upper and lower surfaces (i.e., at z = ±ta /2) of the film be Φ+ and Φ− respectively, so that the potential difference between the two surfaces becomes $Φ = Φ+ − Φ− . Evaluating these two potential boundary conditions with the help of Eq. (23) and simplifying, the potential difference can now be written as ta ∂u0 $Φ = c11 (24) + (c13 + c55 )C4 x − D1 . e31 ∂x The capacitance C of the film segment $x = L2 − L1 and width b, assuming parallel electrodes at the surfaces (z = ±ta /2), can be obtained as L2 + Dz b dx . (25) C= $Φ L1
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The specific cases of boundary constraints are derived in the following subsections. As a special case of the formulation discussed so far, the model of the surface-bonded film on host structure is derived in Section 3.1. The cases of pure tension/compression and pure shear on the embedded film are derived respectively in Sections 3.2 and 3.3. The residual stress in these three cases is not considered. It was reported that the residual stress gradually decreases as the film thickness increases [8]. While studying the effect of tension/compression and shear, we shall consider the thickness of the PVDF film of the order 10 m and PZT film of the order 100 m, so that the assumption of negligible residual stress is fairly accurate. The effect of residual strains/stresses, which is significant in thin film of thickness below the order of m, is discussed in Section 3.4. 3.1. Film on the surface of the host structure The expressions for capacitance and voltage across the electrodes for a surface-bonded film segment is derived here. This requires the elimination of the effect of host structural materials above the upper surface (z = tm + ta /2 in Fig. 1) (retaining the bonding layer, which now represents the insulation layer and electrodes). Also we need to eliminate the traction boundary conditions applied at z = tm + ta /2, since it is now a free surface. The stresses are transferred from the host/substrate material to the film through the lower bonding − and the layer only. We assume that the applied shear stress τxz − displacement u to be linear function of the length coordinate x. This is the simplest possible stress distribution. More complex spatial variation of the boundary layer traction and displacements can be extracted from detail finite element analysis and then they can be applied on the film-substrate interface for calculation of the capacitance and voltage using the present approach. Since the upper surface is free, + = 0, σ + = 0. Equating τ − with the deformation-induced τxz zz xz horizontal shear stress at the lower surface using Eq. (6), we get − (x) τ − (x) τxz 2tm ta 0 − u = u (x) − + ∗ , φ = − xz , c55 G 2 c55 (26) where
L 1 − L − u (x) = u +u − 2 2 2 x − L L + u − u− − , L 2 2 −
− τxz (x) =
1 − L L − τ + τxz − 2 xz 2 2 L x − L − τ − τxz − . + L xz 2 2
Now, Eqs. (15a) and (15b)) can be rewritten as
(27a)
(27b)
C3 = −
c11 ∂u0 , c13 ∂x
C4 = 0.
(28)
Note that the linear variation of the applied shear traction and the horizontal displacement along x have been prescribed using the corresponding stresses and displacements at the two ends of the sensor segment i.e., at x = ±L/2. The capacitance and output voltage of the sensor are obtained by substituting Eqs. (27a) and (27b) in Eqs. (19), (22), (24) and (25), in which 0 ta ∂φ ∂u c13 e31 c33 e31 D1 = c11 − + C3 . (29) − e33 ∂x e33 2 ∂x The explicit form of the capacitance is given by c33 33 be31 (L2 − L1 ) C3 e33 + C= e33 ta (c11 (∂u0 /∂x) − D1 ) 0 ∂u c13 33 ta ∂φ . + + e31 + ∂x 2 ∂x e33
(30)
Numerical results using this analytical solution are shown in Section 4.1. 3.2. Embedded film under tension + and σ − are In this case, only direct tensile stresses σxx xx applied at the two end cross-sections of the film i.e. at x = ±L/2. If the prescribed horizontal displacements at the upper and lower surfaces (z = ±(tm + ta /2)) are assumed as linear variation in x, then similar to Eq. (27a) in the surface-bonded cases (Section 3.1), here we have 1 − L L + − − u u (x) = u (x) = +u − 2 2 2 L x − L − u + −u − . (31a) L 2 2
Assuming axisymmetric configuration about X-axis (see + and τ − respectively at the Fig. 1), the shear tractions τxz xz upper and lower surfaces (z = ±(tm + ta /2)) are L 1 − L + − − τxz (x) = τxz (x) = τ + τxz − 2 xz 2 2 x − L L − + − τxz − τxz . (31b) L 2 2 + = σ − = 0. Substituting Eqs. (31a) and (31b) in Also, σzz zz Eq. (6), L 1 − L 0 − u +u − u = 2 2 2 x − L L − + u −u − L 2 2 tm L − L − + τ + τ − xz 2G∗ xz 2 2 tm x L − L − τxz + ∗ − τxz − , G L 2 2
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
φ = 0.
(32)
In the following analysis, we shall assume that the film is symmetrically embedded in host structure, so that there is a symmetry in shear traction about Z-axis when the host structure is loaded with tensile loading. The boundary layer traction and displacements at the film-substrate interfaces − (−L/2) = are then extracted. In such case, one can write τxz − τxz (L/2). Subsequently, Eqs. (15a) and (15b) can be rewritten as − 1 u (L/2) − u− (−L/2) + − C3 = σ + σxx − 2c11 , 2c13 xx L (33a) C4 =
1 + − σxx − σxx . Lc13
(33b)
Using Eqs. (32)–(33b) in Eqs. (22), (24) and (25), we get − c13 e31 u (L/2) − u− (−L/2) D1 = c11 − e33 L C4 L c33 e31 c33 e31 + c13 + c55 − C3 . − 2 e33 e33
(34)
be31 BC4 F2 (L2 − L1 ) be31 + (BD1 + BC3 F2 2 ta C4 F2 ta C4 F22 + AF2 F3 − F1 F2 − Bc11 F3 ) D1 − c11 F3 − C4 F2 L2 , × log D1 − c11 F3 − C4 F2 L1 c13 33 , e33
F1 = 0,
F2 = c13 + c55 ,
B = e33 +
(35)
c33 33 , e33
F3 =
ta ∂φ ∂u0 + . ∂x 2 ∂x
Numerical results using this analytical solution are shown in Section 4.2. 3.3. Embedded film under shear + is applied at the upper In this case, only shear stress τxz surface (z = tm + ta /2). The lower bonding layer is fixed, i.e. u− = 0, ua (z = −ta /2) = 0. It is assumed that wa = 0 throughout the film segment. A linear variation of shear stress in x is assumed as in Sections 3.2 and 3.3. A linear variation of the upper surface displacement u+ in x is imposed, which is consistent with this assumption. From the displacement continuity at z = ta /2, one can write
u+ =
+ τxz a ta t + u , m G∗ 2
τ+ ua (ta /2) − ua (−ta /2) = xz . ta c55
(36b)
Combining Eqs. (36a) and (36b) and using Eq. (6), we get u0 =
+ ta τxz , 2 c55
φ=
+ τxz . c55
(36c)
Further, Eqs. (15a) and (15b) can be rewritten as + + (−L/2) τxz (L/2) − τxz c11 ta C3 = − , C4 = 0. 2c13 c55 L (36d) Using Eqs. (36c) and (36d) in Eqs. (22), (24) and (25), we get + + (−L/2) (L/2) − τxz ta τxz D1 = c55 L c13 e31 c11 c33 e31 × c11 − + . (37) e33 2c13 e33
3.4. Effect of residual stress
where A = e31 +
and from condition of the shear stress transfer from z = ta /2 to z = −ta /2, one can write
The explicit form of the capacitance is given by Eq. (30). Numerical results using this analytical solution are shown in Section 4.4.
The explicit form of the capacitance is obtained as C=
429
(36a)
In order to include the effect of residual stress in the present model, we consider the expressions for various types of residual stresses as discussed in [8]. In the numerical studies in Section 4.5, we shall consider only three types of residual stresses, namely, the intrinsic, the thermal and the transition stresses. Experimental studies (e.g. [8]) shows that the intrinsic and transition stresses are compressive and thermal residual stresses are tensile in nature. Stresses arising in case of epitaxially grown film, which is called epitaxial stress, is not considered in the present analysis, since it does not contribute to residual stress [19]. Once the total residual stress is estimated by summing up all the individual process-dependent residual stresses, the total equivalent residual strain can be calculated. This total residual strain (εrxx ) can be included in the present model while considering a particular type of constrained boundary. Below we show the formulation for the case of + and σ − applied respectively at in-plane tensile stresses σxx xx the two ends x = ±L/2 of the film as in Section 3.2. The modified forms of Eqs. (33a) including residual strain becomes 1 − σ + + σxx C3 = − 2c11 2c13 xx + u (L/2) − u− (−L/2) (38) × + εrxx . L
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R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437 Table 1 Material properties of the film
Expanded form of Eq. (22) becomes c13 e31 u+ (L/2) − u+ (−L/2) D1 = c11 − e33 L c33 e31 C4 L + c13 + c55 − 2 e33 −
c13 e31 r c33 e31 C3 − ε . e33 e33 xx
(39)
Consequently, the explicit form of the output voltage and the electric displacement can respectively be written from their general form in Eq. (19) and (24) as + u (L/2) − u+ (−L/2) c13 33 r D+ = e + + ε 31 z xx e33 L c33 33 + e33 + (C3 + C4 x), (40) e33 u+ (L/2) − u+ (−L/2) ta c11 $Φ = e31 L + (c13 + c55 )C4 x − D1 .
(41)
∂u0 ta ∂φ + + εrxx . ∂x 2 ∂x
PVDF
PZT
c11 (GPa) c13 (GPa) c33 (GPa) c55 (GPa) Poisson’s ratio (ν) d31 (pC/N) d33 (pC/N) e31 (N/V m) e33 (N/V m) Relative permittivity s33 / 0 at ω = 1 MHz Dissipation factor η Volume resistivity ρ (ohm m)
2.00 0.768 2.00 0.746 0.34 8 15–16 0.02753 0.03615 11 0.018 5 × 1012
69.00 23.664 55.00 26.336 0.31 −175 350 −3.79256 15.1088 1800 0.018 2 × 1010
directly transfered to the sensor film. Also, an additional assumption is made, that is the effective thickness ts of the host structure over which the stress transfer takes place is known. Material properties of the PVDF and PZT film used in the numerical simulations are shown in Table 1. Carbon-epoxy composite (55% fiber volume fraction is used as host structural materials with c11 = 55.1 GPa. The shear modulus of the bonding layer is assumed as G = 1 GPa. The viscoelastic property of this bonding layer is modeled as G∗ = G(1 − iηm ), where the loss factor is ηm = 0.007. The frequency dependent dielectric including the effect of AC loss is given by [20]
Using Eqs. (33b), (38) and (39) in Eqs. (40), (41) and (25), the capacitance of the film segment is given by Eq. (35) with F3 modified as F3 =
Property
(42)
33 = s33 (1 − iη ) −
Numerical results using this analytical results are discussed in Section 4.4.
i , ρr ω
(43)
where s33 is the DC component of the dielectric, η is the AC dielectric loss factor, ρr is the volume resistivity and ω is the AC frequency. In all the numerical simulations, the thickness of the bonding layer tm = 10 m, film width b = 1 mm, film length L = 2 mm and 3 mm respectively for PVDF and PZT, film thickness ta = 10 m and 100 m, respectively for PVDF and PZT. Fig. 2.
4. Numerical results and discussions In this section, we first show the variation in the capacitance and output voltage based on the proposed model (termed as Formulation 1) of the surface-bonded film compared to the results from conventional expression of electrostatics (C = b 33 (L2 − L1 )/ta , termed as Conventional) and another formulation based on simple stress-transfer and strain continuity (termed as Formulation 2). A brief description of Formulation 2 is given in the Appendix. Note that in this simple one-dimensional model, only a constant in-plane tensile stress profile in the host structure is assumed to be
4.1. Surface-bonded film Fig. 3(a) and (b) respectively show the dependence of capacitance of the surface-bonded PVDF film and PZT film on their geometric properties, i.e., film thickness and length. It can be seen that in case of PVDF, the results predicted by dx
Z Host structure
ts
Bonding layer
tm
Film (a)
X
t a /2
s
σxx
σ
xx
s s σ xx + dσ xx
τxz τxz
σ xx + dσxx
(b)
Fig. 2. Schematic diagram of the simplified model of surface-bonded sensor based on stress transfer and strain continuity at the bonding layer.
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431
Formulation 1 Formulation 2 Conventional
Formulation 1 Formulation 2 Conventional
12.5
60 10
Capacitance (nF)
Capacitance (pF)
50 40 30 20 10
7.5 5 2.5 100
50 40
50 Thickness (µ m)
1.5
20 Thickness (µ m) 10
(a)
75
2 30 0.5
1 Length (mm)
25
0.5
(b)
1.5
1
2.5
2
3
Length (mm)
14
14
12
12
10
10
Voltage (V)
Voltage (V)
Fig. 3. Capacitance of (a) PVDF sensor and (b) PZT sensor surface-bonded on carbon-epoxy composite. Formulation 1: present model (Eq. (30); Formulation 2: direct stress transfer and strain continuity (Eq. (A.10)); Conventional: electrostatics.
8 6 4
8 6 4
2
2
0 6 10
0 6 10 0.5
5
10
10
Stress (Pa)
3
(a)
10
0.25 0.5
0 x/L
0.5
5
10
0.25 4
0.25 0
4
10
Stress (Pa)
3
(b)
10
x/L
0.25 0.5
Fig. 4. Output voltage of PVDF sensor surface-bonded on carbon-epoxy composite obtained from (a) Formulation 1 and (b) Formulation 2.
0
0
- 10
- 50
- 20
- 100
- 30 - 40 - 50
- 150 - 200 - 250
- 60
- 300
- 70 3 10
3
10 4
4
0.5
10
10
10
7
10
0.25 0.5
0.25
5
10
0
6
Stress (Pa)
0.5
10
0.25
5
(a)
affect the constrained electro-mechanical behavior and are modeled more accurately through Formulation 1. Fig. 4(a) and (b) respectively shows the variation in the output voltage of the surface-bonded PVDF sensor on carbon-epoxy composite obtained from the present model and Formulation 2. Similarly, Fig. 5(a) and (b) respectively shows the
Voltage (V)
Voltage (V)
the present model is almost identical to the result from conventional model of electrostatics, but differ from Formulation 2. In case of PZT, the present model significantly differs from both the conventional model as well as from the Formulation 2. This can be attributed to the higher stiffness of the PZT film and its higher thickness range that significantly
0
6
x/L
Stress (Pa)
(b)
10
7
10
0.25 0.5
x/L
Fig. 5. Output voltage of PZT sensor surface-bonded on carbon-epoxy composite obtained from (a) Formulation 1 and (b) Formulation 2.
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Formulation 1 Formulation 2 Conventional
12 10 Capacitance (nF)
Capacitance (pF)
60 50 40 30 20
8 6 4 2
10
0 100
50
80
40
2 30
20 Thickness (µ m) 10
(a)
60
1.5
40
1 Length (mm)
0.5
20
Thickness (µ m)
(a)
0
0
1
0.5
1.5
2.5
2
3
Length (mm)
Fig. 6. Variation in the capacitance of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite due to variation in the film thickness and length.
variation in the output voltage of the surface-bonded PZT film on carbon-epoxy composite obtained from the present model and Formulation 2. For all these results using Formu− (−L/2) = 0 and τ − (L/2) is varied as shown in lation 1, τxz xz the figures. It is clear that the low stiffness of the PVDF film produces similar voltage amplitude predicted by the Formulation 1 and the Formulation 2. However, for PZT which has high stiffness, it can be seen from Fig. 5(a) and (b) that there are differences in the voltages. This difference can be attributed to the fact that in Formulation 2, the admissible in-plane tensile stress is constant throughout the film span, which is not the exact case because of certain amount of horizontal shear deformation across the sensor film thickness. This becomes more significant as the film stiffness increases.
Fig. 6(a) and (b) respectively shows the variation in the capacitance of the PVDF film and the PZT film embedded in carbon-epoxy composite for variation in the thickness and length. The classical prediction is that the capacitance becomes infinite as the distance between electrodes becomes zero [21]. Here also we observe the exponential increase in the absolute value of the capacitance with decreasing film thickness. In order to study the effect of tensile stresses on the + at x = L/2 and σ − at x = −L/2 are varcapacitance, σxx xx ied separately by keeping one of them constant at a time. Fig. 7(a) and (b) respectively shows the percentage variations in capacitance for PVDF film and PZT film embedded in carbon-epoxy composite with respect to the conventional estimate. These results indicate that for different combinations of tensile stresses applied at x = ±L/2, the maximum percentage variation in capacitance, compared to the conventional, is nearly 1% for PVDF which has low stiffness and nearly +65 to −40% for PZT which has higher stiffness. Also, it can be seen that when the applied stresses at the ends are nearly equal, the percentage variations in capacitance remain nearly constant for all stress levels.
4.2. Embedded film subjected to tensile stress
1.25
% Variation in Capacitance
% Variation in Capacitance
+, σ− Here, first the effects of in-plane tensile stresses σxx xx and the film length on the capacitance and output voltage are + is studied studied. The effect of transverse normal stress σzz next. In all these cases, the linear spatial variation of the applied tractions are considered as discussed in Section 3.2.
1 0.75 0.5 0.25 0 6 5 4 Log10(σ-xx)
(a)
5
3 2 1
1
2
3
4 Log10(σ+xx)
6
75 60 45 30 15 0 15 30 45 7 6
7
5 4
(b)
Log10(σ-xx)
5
3 2 1
1
2
3
6
7
8
4 + Log10(σxx)
+ and σ − on capacitance of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite. Fig. 7. Effect of tensile stresses σxx xx
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
100 0 - 100 - 200 - 300 - 400 - 500 - 600 - 700 - 800 3 - 10
Voltage (V)
Voltage (V)
800 700 600 500 400 300 200 100 0 100 6 10 0.5
5
10
10
Stress (Pa)
3
(a)
10
0.25
0.25 0
5
Stress (Pa) - 10
x/L
0.5
0.5
4
- 10
0.25 0
4
433
0.25
x/L
0.5
10 6
(b)
600
600
500
500
400
400 Voltage (V)
Voltage (V)
+ at x = +L/2 and (b) σ − at x = −L/2. Fig. 8. Output voltage of PVDF sensor embedded in carbon-epoxy composite under varying tensile stress (a) σxx xx
300 200 100 0
300 200 100 0 - 100
100
- 200 7 - 10
200 7 10 6
6
0.5
10
Stress (Pa)
4
10
3
(a)
10
0.25 0.5
0.5
- 10
0.25
5
10
0.25
5
- 10
0 x/L
Stress (Pa)
0
4
- 10
(b)
10
3
0.25
x/L
0.5
+ at x = +L/2 and (b) σ − at x = −L/2. Fig. 9. Output voltage of PZT sensor embedded in carbon-epoxy composite under varying tensile stress (a) σxx xx
ficient contribute to certain amount of bias field along the film span causing loss of symmetry for the same stress field. 4.3. Embedded film subjected to transverse normal stress In order to study the effect of transverse normal stress, + at the upper surface z = t + t /2 we apply constant σzz m a
0
0
- 50
- 10 - 20
- 100
Voltage (V)
Voltage (V)
Corresponding output voltages for varying stresses at the ends separately can be seen in Fig. 8 for PVDF film and in Fig. 9 for PZT film. It can be seen that for PVDF, higher voltages are developed towards the end segments which are subjected to higher stresses. The response is in direct correlation with the strain induced due to stress field. On the other hand, for PZT, the high stiffness and the polarization coef-
- 150 - 200
- 30 - 40 - 50
- 250
- 60
- 300 3 10
- 70 3 10
(a)
6
10
0.25 0.5
0.25
5
10
0
10
0.5
10
0.25 5
Stress (Pa)
4
0.5
4
10
0
6
x/L
Stress (Pa)
(b)
10
7
10
0.25
x/L
0.5
+ at Fig. 10. Output voltage of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite under varying transverse normal stress σzz z = tm + ta /2.
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
7
7
6
6
5
5
Voltage (V)
Voltage (V)
434
4 3 2
4 3 2
1
1
0 6 10
0 7 10 6
0.5
5
10
4
Stress (Pa)
10
3
(a)
10
0.25 0.5
0.5
10
0.25
0.25
5
10
0 x/L
0
4
Stress (Pa)
10
3
(b)
10
0.25
x/L
0.5
+ at z = +t /2. Fig. 11. Output voltage of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite under shear stress τxz a
while keeping the lower surface restrained (w− = 0). In this case, the expression of capacitance is same as in Eq. (35) + /e . All the other boundary tractions are with F1 = σzz 33 33 zero in the corresponding expressions given in Section 3.2. Fig. 10(a) and (b) respectively shows the output voltage of the PVDF film and the PZT film embedded in carbon-epoxy composite. Since, the applied traction is constant over the film length, the voltage is also constant. However, with the applied stress typically in the order of MPa, higher magnitude of voltage in 10µm PVDF compared to 100 m PZT can be seen.
ical work conjugate are decoupled from the electrical work conjugate i.e. e51 = 0. As it can be noted from Section 3.3 that the above simplification has made the present approach adequate to deal with arbitrary shear traction distribution + and τ − . Following further simplification of the analytiτxz xz cal solution, the effect of linear shear traction on the voltage is studied. Although, the mechanical and electrical work conjugates are decoupled, the boundary constraints on the embedded film indeed produce a voltage, which is shown in Fig. 11(a) and (b) respectively for PVDF and PZT films. Here, the applied shear stress is varying linearly from x = −L/2 to x = +L/2. This shear stress distribution results in a constant stress gradient along the film-substrate interface. From Eqs. (24), (36d) and (37), it is clear that the output voltage is a function of shear stress gradient. However, Fig. 11 shows that due to decoupled shear and electric field, the magnitude of voltage is quite less as compared to that in cases of equivalent order of applied tensile stress (Figs. 8 and 9) and applied transverse normal stress (Fig. 10).
4.4. Embedded film subjected to shear stress In Section 3.3, a solution has been obtained, where the embedded film is subjected to pure shear stress at the upper and lower interfaces between the bonding layer and the host structure. This model is based on the constitutive relation, where unlike anisotropic piezoelectric material, the mechan-
20 18 16
14
Capacitance (nF)
12 Voltage (V)
10 8 6 4 2 0 2 7 10
With residual stress Conventional Without residual stress
14 12 10 8 6 4
6
10
5
10 Applied tensile (a) stress (Pa)
4
10
3
10
0.5
0.25
0.25
0 x/L
2
0.5
0 3
(b)
3.5
4
4.5
5
5.5
6
6.5
7
+
Applied tensile stress (log[σxx])
Fig. 12. (a) Output voltage of 1 mum PZT film under varying applied tensile stress along with compressive residual stress (shown by thick solid lines) and without the residual stress (shown by thin solid lines). (b) Effect of residual stress on capacitance of 1 m PZT film at various mechanical tensile stress levels.
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
4.5. Effect of residual stress Here we show the comparative response of a 1 m PZT film deposited on Pt/Ti/Si(0 0 1), with and without the residual stress. The explicit form of the output voltage and the capacitance have been obtained in Section 3.4. The values of residual stress components are calculated on the basis of formulae given in [8]. As discussed in Section 3.4, the values of intrinsic, thermal and transition stress are estimated as 117.2, 217 and 110.2 MPa, respectively. Therefore the total residual stress is sum of these three values, which is found to be 10.2 MPa compressive. In the present calcula+ and σ − are applied tions, the mechanical tensile stress σxx xx at the two ends x = ±L/2 are made equal. The compressive residual stress estimated above is applied through the equivalent residual strain using Eq. (41). Fig. 12(a) shows the the variation in the voltage at various locations of the film under varying tensile stress. The constant tensile stress over the film length is increased from 1 kPa to 10 MPa. It can be observed from Fig. 12(a) that the voltage in presence of the estimated compressive residual stress is approximately 4 V, which is higher than the voltage without the residual stress. This can be attributed to some amount of residual charge which is already stored on the film surface due to induced residual strain. It is not only the voltage that is affected by the residual stress, but also the capacitance, which is clear from Fig. 12(b). From this figure, it can be observed that the capacitance including the effect of residual stress is nearly 12 nF lesser than the capacitance without the residual stress. However, when the applied tensile stress is in the order of 1–10 MPa, the effect of compressive residual stress is nullified and the capacitance drops showing a very small amount of transduction caused by mechanical deformation. In contrast, the conventional estimate is independent of the effect of any applied or residual stress. According to the definition of the capacitive sensor, it is the capacity of the sensor material to accumulate the electrical charge. The present formulation and the above numerical results clearly show that the equivalent capacitance of a piezoelectric film is actually dependent on the constrained boundary, applied mechanical load and also the residual stress.
5. Conclusions A new analytical model has been reported to analyze the performance of piezoelectric thin film sensors, surface-bonded or embedded in composite material system. As an improvement over the existing models, the proposed model includes several non-classical effects, such as viscoelastic property of the bonding layer, AC dielectric loss and frequency-dependent effect of the resistivity of the film materials, and more importantly, the effect of constrained boundary. Governing equations are solved exactly by considering first-order shear deformation. General expression for the electric charge displacements and potential
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difference are obtained. The boundary constraints can be calculated a-priori using appropriate analytical solution, or in more detail, by boundary layer stress and displacement extraction from finite element analysis of the host structure. The analytical solution shows significant complexity of the functional dependence of various parameters, such as, layer geometry, combination of material properties and stress. Four cases are solved and studied numerically, which are, (1) when the film is on the surface of the host structure, (2) when the embedded film is subjected to in-plane tensile stress, (3) when the embedded film is subjected to transverse normal stress and (4) when the embedded film is subjected to pure shear. In all these cases, the performance of PVDF film and PZT film is studied for a wide range of applied stress. In case of surface-bonded film, the results are compared with existing models. For the cases of embedded film, the variations in the voltage due to change in the stresses are found to be significant, which are otherwise difficult to simulate using exiting model. The results indicate significant trade-off that may be necessary in design optimization of piezoelectric thin film device for various applications. Also, a clear numerical illustration is made to show the effect of residual stress on the capacitance and voltage. This is an important effect to consider, which is a common issue in most of the MEMS and thin-film fabrication and packaging processes. The proposed model can be useful while dealing with complex mechanical loading. The analytical solution can also be useful in optimizing the capacitive performance. Additional complexities, such as the nano-scale effects, may be possible to incorporated in the present model.
Appendix A. A simple model based on stress transfer and strain continuity A piezoelectric film bonded to the surface of the host structure is considered. A constant in-plane tensile stress profile in the host layer is assumed to be directly transfered to the film along with the condition of strain continuity at the film-substrate interface. The free-body diagram is shown in Fig. 2. The stress transfer mechanism is based on shear-lag model [20]. The superscript s indicates the variables in the host/ substrate structure. The superscript a indicates the variables in the film. As shown in Fig. 2, ts , tm and ta /2 are respectively the thicknesses of the host structure (an equivalent thickness for effective stress transfer), the thickness of the bonding layer and the thickness of the film. Considering a span-wise segment of infinitesimal length dx of the film along with the bonding layer, the equilibrium equations can be written as dσxx ta b + 2b dxτxz = 0,
s dσxx ts b − b dxτxz = 0.
(A.1)
where τxz is the shear stress transferred from host structure to the film through the bonding layer. From the shear-lag
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model, we have us − u a τxz = G∗ . tm
(A.2)
Differentiating Eq. (A.1) thrice with respect to x and subs = cs εs , yields after simplistituting σxx = c11 εxx and σxx 11 xx fication, 2 s d4 εsxx 2 d εxx − Γ = 0, 0 dx4 dx2
where Γ02
G∗ = s c11 ts tm
2 d4 εxx 2 d εxx − Γ = 0, 0 dx4 dx2
s t +c t 2c11 s 11 a c11 ta
(A.3)
.
(A.4)
The general solution for strains in Eq. (A.3) is given by εxx = C1 + C2 x + C3 sinh (Γ0 x) + C4 cosh(Γ0 x), εsxx = C1 + C2 x −
c11 ta s t (C3 sinh (Γ0 x) + C4 cosh(Γ0 x)) . 2c11 s (A.5)
Boundary conditions are now applied separately on the host structure and on the film, which are σs L εsxx (x) = sxx , σxx ± = 0. (A.6) c11 2 The unknown constants are then expressed as s σxx s (1 + (c t /2cs t )) , c11 11 a 11 s C1 C4 = − . cosh(Γ0 (L/2))
C1 =
C2 = 0,
C3 = 0, (A.7)
For pure sensory transduction, i.e., assuming all the stresses are transformed into electric potential and dissipated through the circuit, σxx = c11 εxx − e31
$Φ = 0, ta
Dz = e31 εxx + 33
$Φ . ta (A.8)
Surface charge accumulated over the film segment of width b and span (x = −L/2, +L/2) is given by 2 e31 bc11 σxx Q= + 33 s s t )) e31 c11 (1 + (c11 ta /2c11 c11 s +L/2 cosh(Γ0 x) 1− × (A.9) L dx. cosh Γ0 2 −L/2 Equating the deformation-induced electric power, the equivalent capacitance of the film segment L is obtained as 2 e31 b 4 sinh (Γ0 (L/2)) C= L− + 33 ta c11 cosh(Γ0 (L/2))Γ0 2 sinh (Γ0 L) 1 + + 2L . (A.10) Γ0 4cosh2 (Γ0 (L/2))
In absence of any piezoelectric effect, the conventional form of electrostatic capacitance can be clearly seen from Eq. (A.10). References [1] S.D. Senturia, Microsystems Design, Kluwer Academic Publishers, 2000. [2] M. Yocum, H. Abramovich, A. Grunwald, S. Mall, Fully reversed electromechanical fatigue behavior of composite laminate with embedded piezoelectric actuator/sensor, Smart Mater. Struct. 12 (2003) 1–9. [3] C.V.S. Sastry, D. Roy Mahapatra, S. Gopalakrishnan, T.S. Ramamurthy, Effect of actuation phase on the SIF in delaminated composite with embedded piezoelectric layers, Smart Mater. Struct. 12 (2003) 602–611. [4] C.V.S. Sastry, D. Roy Mahapatra, S. Gopalakrishnan, T.S. Ramamurthy, Distributed sensing of static and dynamic fracture in self-sensing piezoelectric composite: finite element simulation, Intell. Mater. Syst. Struct. 15 (2004) 339–354. [5] D.F. Bahr, J.S. Robach, J.S. Wright, L.F. Francis, W.W. Gerberich, Mechanical deformation of PZT thin films for MEMS applications, Mater. Sci. Eng. A 259 (1999) 126–131. [6] D.F. Diao, K. Kato, K. Hokkirigawa, Fracture mechanisms of ceramic coatings in indentation, ASME J. Tribol. 116 (1994) 860–869. [7] A. Steegen, K. Maex, Silicide-induced stress in Si: origin and consequences for MOS technologies, Mater. Sci. Eng. R38 (2002) 1–53. [8] Y.C. Zhou, Z.Y. Yang, X.J. Zheng, Residual Stress in PZT thin films prepared by pulsed laser deposition, Surf. Coatings Technol. 162 (2003) 202–211. [9] T.Y. Zhang, L.Q. Chen, R. Fu, Measurements of residual stresses in thin films deposited on silicon wafers by indentation fracture, Acta Mater. 47 (14) (1999) 3869–3878. [10] J.W. Hutchinson, Z. Suo, Mixed mode cracking in layered materials, Advances in Applied Mechanics, vol. 29, 1992, Academic Press, New York, pp. 63–191. [11] A.G. Evans, J.W. Hutchinson, The thermomechanical integrity of thin films and multilayers, Acta Metall. Mater. 43 (7) (1995) 2507–2530. [12] W.D. Nix, Mechanical properties of thin films, Metall. Trans. 20 (1988) 2217–2245. [13] E. Sushir, Interfacial stresses in bimetal thermostats, ASME J. Appl. Mech. 56 (1989) 595–600. [14] E. Sushir, Analysis of interfacial thermal stresses in a trimaterial assembly, J. Appl. Phys. 89 (7) (2001) 3685–3694. [15] T.-C. Chen, H.-C. Wu, C.-L. Lin, Longitudinal anisotropic stress and deformation in multilayered film heterostructures due to lattice misfit, J. Cryst. Growth 249 (2003) 44–58. [16] G. Schrag, G. Zelder, H. Kapels, G. Wachutka, Numerical and experimental analysis of distributed electromechanical parasitics in the calibration of a fully BiCMOS-integrated capacitive pressure sensor, Sens. Actuators A (physical) 76 (1999) 19–25. [17] N. Mokni, F. Sidoroff, A. Danescu, A one-dimensional viscoelastic model for lateral relaxation in thin films, Comput. Mater. Sci. 26 (2003) 56–60. [18] T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, New York, 1990. [19] C.M. Foster, Z. Li, M. Buckett, D. Miller, P.M. Baldo, L.E. Rehn, D.R. Bai, D. Guo, H. You, K.L. Merkle, Substrate effects on the structure of epitaxial PbTiO3 thin films prepared on MgO, LaAlO3 and SrTiO3 by metalorganic chemical vapor deposition, J. Appl. Phys. 78 (4) (1995) 2607–2622. [20] A. Pizzochero, Residual actuation and stiffness properties of piezoelectric composites: theory and experiment, M.S. Thesis, MIT, 1998. [21] C. Trimarco, The electric capacitance of a rigid dielectric structure, Int. J. Solids Struct. 29 (13) (1992) 1647–1655.
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Biographies R. Ali is currently working at the aerospace engineering department, Indian Institute of Science (IISc), Bangalore, as research assistant in the field of modeling of thin films for MEMS and SHM applications. Prior to this, he worked at Advanced Composites Division of National Aerospace Laboratories, Bangalore, India, as Research Assistant, in the field of finite element modeling of shape memory alloy wires embedded in composites for actuators in aerospace applications. He received his ME degree in Aerospace Engg. from IISc, in January 2002. His research interests are in smart structures, composites, non-destructive testing and structural health monitoring (SHM) of aerospace and industrial structures. D. Roy Mahapatra received his BE degree in Civil Engineering from Jadavpur University, Calcutta, India, in 1998 and received his PhD in Aerospace Engg. from Indian Institute of Science (IISc), Bangalore in 2004. Currently, he is working as a postdoctoral fellow in the department of Aerospace Engg., IISc. His current areas of research are on the
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development of finite element techniques for analysis of vibration and wave propagation in composite structures, design of functional composite for vibration sensing and control applications, design of micro and nano systems, interaction of wave with materials, shock wave and hypersonics. S. Gopalakrishnan received his BE degree in Civil Engineering from the Bangalore University, India in 1984 and MTech degree in Engineering Mechanics from the Indian Institute of Technology (IIT), Chennai in 1987. He received his PhD. from the School of Aeronautics and Astronautics, Purdue University, West Lafeyette, USA in 1992. Later, he was a post doctoral fellow in the School of Mechanical Engineering at Georgia Tech., Atlanta, USA, from January 1993 to July 1994. He then became an Assistant Professor in the Department of Civil Engineering, IIT, Chennai. He joined the Department of Aerospace Engineering at the Indian Institute of Science (IISc), Bangalore as an Assistant Professor in November 1997. He became an associate professor in the same department in 2003 and continuing in this position. His main areas of interest are structural dynamic, wave propagation, smart structures and FEA.
An analytical model of constrained piezoelectric thin film sensors R. Ali, D. Roy Mahapatra, S. Gopalakrishnan∗ Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India Received 6 October 2003; received in revised form 28 April 2004; accepted 2 May 2004 Available online 8 July 2004
Abstract A new analytical model based on 2D electro-mechanical continuum model is developed to analyze the performance of constrained piezoelectric thin film sensors, surface-bonded or embedded in composite material system. Several non-classical effects, such as viscoelastic property of the bonding layer, AC dielectric loss, frequency-dependent effect of the resistivity of the film materials, and most importantly, the effect of constrained boundary on the capacitive performance are considered. The model is validated by comparing the results from electrostatics as well as a simplified model based on direct stress transfer and strain continuity at the film-substrate interface. Analytical solutions for specific cases of boundary constraints on the film, such as in-plane tensile stress, transverse normal stress and horizontal shear stress are reported. Numerical studies on the effect of these stresses on PVDF and PZT films with carbon-epoxy host structure are carried out. Effect of process-induced residual stress is also studied. The results show significant complexity, which is otherwise intractable using existing simplified approaches. © 2004 Elsevier B.V. All rights reserved. Keywords: Constrained piezoelectric thin film; MEMS; Capacitance; Residual stress; Voltage
1. Introduction This paper discusses a new analytical model of constrained piezoelectric thin film sensors, surface-bonded or embedded in a composite structure. The approach is one step improvement over the already prevailing analysis and design methodology for conventional micro-electro-mechanical transducers [1]. The state of boundary constraints on piezoelectric film sensor is important to account in many applications. For example, in structural health monitoring (SHM) applications, the identification of hidden structural defects and any damage process taking place at sub-surface and/or in deep inter-laminar regions in composites using film sensor are of great importance [2–4]. Conventional strain gauges and MEMS sensor arrays are being thought of performing the damage detection task. Another example is the electro-active membrane type adaptive structures for shape and vibration control applications, where the understanding of the effect of various material and geometric parameters on the electro-mechanical coupling is essential. One of the major concerns for MEMS is its relatively poor reliability caused by the delamination, brittle fracture and
∗ Corresponding author. Tel.: +91 80 22933019; fax: +91 80 23600134. E-mail address: [email protected] (S. Gopalakrishnan).
0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.05.028
fatigue degradation of multi-layer thin film structures [5,6]. However, the task of sensing the deformation pattern (e.g. damage detection) can be performed best if the film of large planer dimension with segmented electrode arrays can be modeled, designed and fabricated. While validating such design concept, it is essential to incorporate additional effects, such as, the electro-mechanical coupling in the sensor structure, the dissipation and the electrical resistivity of the sensor material, AC frequency, viscoelastic loss in the adhesive matrix, as against the conventional design [1] where all these effects are not taken into consideration. A major problem with adhesively bonded wafer, or more significantly with deposited thin film, is the control of the state of mechanical stress. Various sources of stress may be discerned when a thin film is deposited or formed by reaction on a substrate [7]. This stress, namely the residual stress, is induced inevitably in thin film due to structural change and thermal misfit between the film and the substrate and the change from high to low temperature during the process [8]. The residual stress significantly affects the service-life of the film, and hence of the whole device itself [8,9]. The residual tensile stress may cause surface crack or tunneling crack in the film [10,11]. The residual compressive stress may cause the film to delaminate from the substrate. The residual stress mainly comprises of Epitaxial, Intrinsic, Transformation and Thermal stresses [7,8]. The strains as-
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
sociated with each of these stresses can be obtained as discussed in the works reported in [12]. With the introduction of simplified interface compliance, both the thermal, as well as the interfacial stresses between the film and the substrate can be estimated [13,14]. In order to design and fabricate better transducer devices with higher reliability, the effect of thickness and length of the piezoelectric film, substrate properties and misfit strain on the distributions of transverse and in-plane stresses must be precisely known. Since the lattice of the thin film and substrate shall be matched with each other during film growth, a misfit strain will be introduced at the interface [15]. The model proposed in this paper will be capable of handling such pre-estimated residual stress and misfit strains while analyzing the capacitive performance of the device. Most of the analytical and finite element analysis technique for coupled electro-mechanical simulations reported in literature are based on iteration over mechanical and electrical equivalent of stress field to achieve optimum capacitance and voltage (e.g. see [16]). In recent time, there has been increasing effort in developing analytical technique to study the constrained effect on thin films [15,17]. Similar problems in distributed transducer application is also expected to play important role. Hence, a systematic analytical model of the coupled electro-mechanical field appears advantageous. In the present paper, we develop a two dimensional electro-mechanical continuum field model to analyze the effects of boundary constraints on the transduction performance of thin as well as thick piezoelectric films. The analytical model proposed in this paper accommodates plane stress or plane strain constitutive properties of piezoelectric layer across the layer thickness, AC frequency dependent dielectric properties, viscoelastic bonding layer and homogeneous substrate material properties. Experimentally observed residual stress field can also be incorporated towards achieving a better design methodology. Closed-form analytical solutions for the capacitance and voltage are derived under various boundary constraints. Numerical results are presented by considering a range of layer thicknesses, lengths and stresses on PVDF and PZT films. To begin with the numerical studies, a database was collected for different sets of materials used for the host structure and the film. Among these sets, results are presented for carbon-epoxy composite as host structural material, and PVDF or PZT as film material. Parametric variation of capacitance and sensor voltage are studied in detail. The results are compared with those used in conventional design. The voltage variation over the length of the film is shown for several cases of direct tension, and surface tractions. The effect of increasing stress level on the capacitance, within the limit of the structural failure, is also presented. The surface-bonded film performance is simulated as a special case of embedded film. Its performance is compared with that of a film based on a very simplistic approach of stress transfer and strain continuity at the surface of the host structure.
425
2. Piezoelectric constitutive model A plane stress case in the plane (X, Z) of the film embedded in host structure is considered here. Z indicates the thickness coordinate of the film. The schematic free body diagram of the sensor is shown in Fig. 1. Under plane stress condition, the 3D constitutive model of the bulk piezoelectric materials [18] can be reduced to 2D in (X, Z) plane by imposing the out-of-plane stresses σyy = 0, τxy = 0, τyz = 0 and the electric charge displacements in the plane (X, Z), Dy = 0 and in the plane (Y, Z), Dx = 0. As a result, the 2D constitutive model can be expressed as a εaxx σ c c 0 −e 11 13 31 xx a σa c εzz c 0 −e 13 33 33 zz a = (1) γ a xz , 0 τxz 0 c55 0 ∂Φ Dz e31 e33 0 33 ∂z where the superscript a indicates sensor layer, cij are the plane-stress reduced elastic moduli, eij are the plane-stress reduced electro-mechanical coupling coefficients, 33 is the dielectric constant of the piezoelectric film, Dz is the electric charge displacement causing deformation-induced charge accumulation on the film surfaces parallel to (X, Y ) plane. The associated conjugate electric field is ∂Φ/∂z, where Φ is the electric potential across the electrodes parallel to the (X, Y ) plane. Here the strain εxx comprises of the strain due to elastic deformation εdxx and the residual strain εrxx . Therefore, the total strain along X-axis is given by εaxx = εdxx +εrxx . Note that in Eq. (1), the coupling between the shear strain and the electric field is neglected. Such constitutive model is applicable to transversely polarized piezoelectric film. The governing equations can be expressed as a ∂τ a ∂σxx + xz = 0, ∂x ∂z
(2a)
a ∂τ a ∂σzz + xz = 0, ∂z ∂x
(2b)
∂Dz = 0. ∂z
(2c)
Here, Eq. (2c) is the reduced form of one of the Maxwell’s equation ∇×D = ρ for material medium including the effect of polarization, where ρ is the total charge density. Under the Z
+ σzz
τ+xz tm σxx
ta
X
Sensor layer
tm
τ xz
Bonding layer Substrate
σ zz
Fig. 1. Schematic diagram of the embedded sensor.
σ+xx
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R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
closed-circuit condition, the accumulated charge on the film surfaces, which is due to the deformation, dissipates across the interface electronic circuit. Hence, for simplicity and by assuming negligible residual charge, one has Eq. (2c). Using Eq. (1) in Eqs. (2a)–(2c),
G∗ is the complex shear modulus of the bonding layer by taking into account its viscoelastic property. Assuming the residual strain as constant over the film length, the strains can be expressed in terms of the displacements as
∂εa ∂γ a ∂εa ∂2 Φ c11 xx + c13 zz + c55 xz − e31 = 0, ∂x ∂x ∂z ∂x∂z
(3a)
εaxx =
∂εa ∂γ a ∂εaxx ∂2 Φ + c33 zz + c55 xz − e33 2 = 0, ∂z ∂z ∂x ∂z
(3b)
∂εa ∂εaxx ∂2 Φ + e33 zz + 33 2 = 0. ∂z ∂z ∂z
(3c)
c13 e31
Let ua and wa be the displacements in the fim along the X-axis and Z-axis respectively. Since the film thickness is much smaller compared to its length, we can assume the inplane and bending deformations to be dominant compared to the transverse shear deformation. Assuming first order shear deformation across the film thickness, ua can be given by (5)
where u0 is the film mid-plane (z = 0) displacement of the sensor. φ is the rotation of the vertical cross-section about Y -axis. Eq. (5) can further be approximated in terms of the applied displacements and traction on the upper and lower surfaces of the sensor embedded in host substrate through the bonding layer (see Fig. 1), which yields u+ + u− tm + − − (τ + τxz ), 2 2G∗ xz u+ − u− tm + − φ= − ∗ (τxz − τxz ), ta G ta
u0 =
(6)
where u+ and u− are respectively the horizontal displacements at the upper and the lower surfaces (z = ±(tm +ta /2)) + and τ − are respectively the horizontal of the sensor. τxz xz shear stresses at the upper and the lower surfaces (z = ±(tm + ta /2)) of the sensor. ta and tm are respectively the thicknesses of the film and the bonding layer. Here, the bonding layer is assumed to be an equivalent single layer including the electrodes, insulation layer, adhesive layer etc.
∂wa , ∂z
a =φ+ γxz
∂ 2 wa ∂ 2 wa ∂φ + k + (c55 + k1 ) = 0, 2 2 2 ∂x ∂x ∂z
where e31 e33 k1 = c13 + , 33
The steps involved in obtaining the proposed analytical solution for the constrained piezoelectric film shown in Fig. 1 are discussed below. Eliminating Φ from Eq. (3b) using Eq. (3c), one can write a ∂γxz e31 e33 a ∂ c55 εxx + c13 + ∂x ∂z 33 2 e33 + c33 + εa = 0 (4) 33 zz
εazz =
∂wa . ∂x (7)
Substituting the above, Eq. (4) can be simplified as c55
3. Analytical solution
ua = u0 + zφ,
∂u0 ∂φ +z + εrxx , ∂x ∂x
e2 k2 = c33 + 33 33
(8) .
(9)
Eq. (8) is the governing equation for transverse bending and shear in the film. Observing this equation, wa can be expressed as a function of φ, which is given by c55 + k1 wa = − φ dx + C1 x + C2 + z(C3 + C4 x), c55 (10) where C1 , C2 , C3 , C4 are all constants of integration. Subsequently, ∂wa = −αφ + C1 + C4 z, ∂x where k1 α= 1+ . c55
∂wa = C3 + C4 x, ∂z
(11)
(12)
The objective is now to eliminate the constants Cj with the help of appropriate boundary conditions at the film edges, the upper and the lower surfaces. Since we have assumed that the transduction takes place due to the deformation induced electric potential Φ across the parallel segmented electrodes at the upper and lower surfaces of the film, there are no electrodes or material discontinuities at the two vertical cross-sections of the film at x = ±L/2, where L is the film length. On these two vertical surfaces, the normal traction is given by 0 ∂u ∂φ a σxx +z + εrxx + c13 (C3 + C4 x) . = c11 (13) ∂x ∂x + Considering the cross-sectional average normal stresses σxx − and σxx at the mid-points of the vertical surfaces of the film respectively at x = ±L/2, z = 0 (see Fig. 1), we get two boundary conditions 0 L ∂u + c11 + εrxx = σxx + c13 C3 + C4 , (14a) ∂x 2 x=L/2 0 L ∂u r − c11 + c13 C3 − C4 . (14b) + εxx = σxx ∂x 2 x=−L/2
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
From the above two equations, one can evaluate C3 and C4 as 1 ∂u0 + − σxx + σxx − c11 C3 = 2c13 ∂x x=L/2 ∂u0 r + , (15a) + 2εxx ∂x x=−L/2 1 ∂u0 + − σ − σxx − c11 C4 = Lc13 xx ∂x x=L/2 ∂u0 − . ∂x x=−L/2
(15b)
To evaluate the rest of the two constants C1 and C2 , we assume wa = 0 at the lower end corners i.e. at x = ±L/2, z = −ta /2. This is a typical restraint to model the four-point bending of a film and also can be used to model (1) pure horizontal shear, (2) thin film on a stiff substrate subjected to surface horizontal traction and (3) thickness-wise symmetric embedded film/wafer and host structure subjected to pure tension/compression. Substituting these two displacement boundary conditions in Eq. (10) and simplifying, we get C4 ta α C1 = , − φ dx + φ dx 2 L x=L/2 x=−L/2 (16a) α C3 ta + C2 = 2 2
φ dx
x=L/2
+
φ dx
x=−L/2
. (16b)
Now, the displacement field can be described explicitly with the help of Eqs. (6) and (10). Further, they can be used to explicitly describe the strain field in Eq. (7). Simplifying the electric displacement Dz in Eq. (1) with the help of the explicit form of the strain field and evaluating at z = ta /2, we get 0 ∂u ta ∂φ r D+ = e + + ε 31 z xx + e33 (C3 + C4 x) ∂x 2 ∂x ∂Φ + 33 . (17) ∂z z=ta /2 In order to eliminate the electric potential Φ from Eq. (17), we use one additional boundary condition, that is the normal + (x) applied on the upper surface (z = t /2) of traction σzz a the film. With the help of Eq. (1) one can write 0 ∂Φ 1 ∂u ta ∂φ + r = −σzz (x) + c13 + + εxx ∂z z=ta /2 e33 ∂x 2 ∂x c33 + (C3 + C4 x). (18) e33
Substituting back in Eq. (17), we get 0 ∂u c13 33 ta ∂φ r = e + + + ε D+ 31 z xx e33 ∂x 2 ∂x + (x) 33 σzz c33 33 + e33 + (C3 + C4 x) − . e33 e33
427
(19)
The next objective is to obtain the explicit form of the potential difference between the upper and lower surface of the film, so that the capacitance of the film under various types of boundary constraints can be estimated. In order to do this, we shall reconsider the governing Eqs. (3a)–(3c). Out of these three equations, since Eqs. (3b) and (3c) have already been used, we shall now use Eq. (3a) to derive the potential difference. Using the expressions for strains given by Eq. (7) and substituting in Eq. (3a), we get 2 0 ∂ u ∂2 φ ∂2 Φ c11 + C + z (c + c ) − e = 0. 4 13 31 55 ∂x∂z ∂x2 ∂x2 (20) Integrating Eq. (20) once with respect to x, 0 ∂u ∂φ ∂Φ c11 +z + (c13 + c55 )C4 x − e31 = D1 , ∂x ∂x ∂z (21) where D1 is the constant of integration. D1 can be eliminated by evaluating Eq. (21) at x = L/2, z = ta /2 as 0 ∂u c13 e31 ta ∂φ D1 = c11 − + e33 ∂x 2 ∂x x=L/2 C4 L c33 e31 + c13 + c55 − 2 e33 e31 + c13 e31 r (σ (x)|x=L/2 − c33 C3 ) − + ε . (22) e33 zz e33 xx Integrating Eq. (21) once with respect to z, ∂u0 z2 ∂φ c11 z + + (c13 + c55 )C4 xz − e31 Φ ∂x 2 ∂x = D1 z + D2 ,
(23)
where D2 is another constant of integration. Let the electric potential at the upper and lower surfaces (i.e., at z = ±ta /2) of the film be Φ+ and Φ− respectively, so that the potential difference between the two surfaces becomes $Φ = Φ+ − Φ− . Evaluating these two potential boundary conditions with the help of Eq. (23) and simplifying, the potential difference can now be written as ta ∂u0 $Φ = c11 (24) + (c13 + c55 )C4 x − D1 . e31 ∂x The capacitance C of the film segment $x = L2 − L1 and width b, assuming parallel electrodes at the surfaces (z = ±ta /2), can be obtained as L2 + Dz b dx . (25) C= $Φ L1
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R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
The specific cases of boundary constraints are derived in the following subsections. As a special case of the formulation discussed so far, the model of the surface-bonded film on host structure is derived in Section 3.1. The cases of pure tension/compression and pure shear on the embedded film are derived respectively in Sections 3.2 and 3.3. The residual stress in these three cases is not considered. It was reported that the residual stress gradually decreases as the film thickness increases [8]. While studying the effect of tension/compression and shear, we shall consider the thickness of the PVDF film of the order 10 m and PZT film of the order 100 m, so that the assumption of negligible residual stress is fairly accurate. The effect of residual strains/stresses, which is significant in thin film of thickness below the order of m, is discussed in Section 3.4. 3.1. Film on the surface of the host structure The expressions for capacitance and voltage across the electrodes for a surface-bonded film segment is derived here. This requires the elimination of the effect of host structural materials above the upper surface (z = tm + ta /2 in Fig. 1) (retaining the bonding layer, which now represents the insulation layer and electrodes). Also we need to eliminate the traction boundary conditions applied at z = tm + ta /2, since it is now a free surface. The stresses are transferred from the host/substrate material to the film through the lower bonding − and the layer only. We assume that the applied shear stress τxz − displacement u to be linear function of the length coordinate x. This is the simplest possible stress distribution. More complex spatial variation of the boundary layer traction and displacements can be extracted from detail finite element analysis and then they can be applied on the film-substrate interface for calculation of the capacitance and voltage using the present approach. Since the upper surface is free, + = 0, σ + = 0. Equating τ − with the deformation-induced τxz zz xz horizontal shear stress at the lower surface using Eq. (6), we get − (x) τ − (x) τxz 2tm ta 0 − u = u (x) − + ∗ , φ = − xz , c55 G 2 c55 (26) where
L 1 − L − u (x) = u +u − 2 2 2 x − L L + u − u− − , L 2 2 −
− τxz (x) =
1 − L L − τ + τxz − 2 xz 2 2 L x − L − τ − τxz − . + L xz 2 2
Now, Eqs. (15a) and (15b)) can be rewritten as
(27a)
(27b)
C3 = −
c11 ∂u0 , c13 ∂x
C4 = 0.
(28)
Note that the linear variation of the applied shear traction and the horizontal displacement along x have been prescribed using the corresponding stresses and displacements at the two ends of the sensor segment i.e., at x = ±L/2. The capacitance and output voltage of the sensor are obtained by substituting Eqs. (27a) and (27b) in Eqs. (19), (22), (24) and (25), in which 0 ta ∂φ ∂u c13 e31 c33 e31 D1 = c11 − + C3 . (29) − e33 ∂x e33 2 ∂x The explicit form of the capacitance is given by c33 33 be31 (L2 − L1 ) C3 e33 + C= e33 ta (c11 (∂u0 /∂x) − D1 ) 0 ∂u c13 33 ta ∂φ . + + e31 + ∂x 2 ∂x e33
(30)
Numerical results using this analytical solution are shown in Section 4.1. 3.2. Embedded film under tension + and σ − are In this case, only direct tensile stresses σxx xx applied at the two end cross-sections of the film i.e. at x = ±L/2. If the prescribed horizontal displacements at the upper and lower surfaces (z = ±(tm + ta /2)) are assumed as linear variation in x, then similar to Eq. (27a) in the surface-bonded cases (Section 3.1), here we have 1 − L L + − − u u (x) = u (x) = +u − 2 2 2 L x − L − u + −u − . (31a) L 2 2
Assuming axisymmetric configuration about X-axis (see + and τ − respectively at the Fig. 1), the shear tractions τxz xz upper and lower surfaces (z = ±(tm + ta /2)) are L 1 − L + − − τxz (x) = τxz (x) = τ + τxz − 2 xz 2 2 x − L L − + − τxz − τxz . (31b) L 2 2 + = σ − = 0. Substituting Eqs. (31a) and (31b) in Also, σzz zz Eq. (6), L 1 − L 0 − u +u − u = 2 2 2 x − L L − + u −u − L 2 2 tm L − L − + τ + τ − xz 2G∗ xz 2 2 tm x L − L − τxz + ∗ − τxz − , G L 2 2
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
φ = 0.
(32)
In the following analysis, we shall assume that the film is symmetrically embedded in host structure, so that there is a symmetry in shear traction about Z-axis when the host structure is loaded with tensile loading. The boundary layer traction and displacements at the film-substrate interfaces − (−L/2) = are then extracted. In such case, one can write τxz − τxz (L/2). Subsequently, Eqs. (15a) and (15b) can be rewritten as − 1 u (L/2) − u− (−L/2) + − C3 = σ + σxx − 2c11 , 2c13 xx L (33a) C4 =
1 + − σxx − σxx . Lc13
(33b)
Using Eqs. (32)–(33b) in Eqs. (22), (24) and (25), we get − c13 e31 u (L/2) − u− (−L/2) D1 = c11 − e33 L C4 L c33 e31 c33 e31 + c13 + c55 − C3 . − 2 e33 e33
(34)
be31 BC4 F2 (L2 − L1 ) be31 + (BD1 + BC3 F2 2 ta C4 F2 ta C4 F22 + AF2 F3 − F1 F2 − Bc11 F3 ) D1 − c11 F3 − C4 F2 L2 , × log D1 − c11 F3 − C4 F2 L1 c13 33 , e33
F1 = 0,
F2 = c13 + c55 ,
B = e33 +
(35)
c33 33 , e33
F3 =
ta ∂φ ∂u0 + . ∂x 2 ∂x
Numerical results using this analytical solution are shown in Section 4.2. 3.3. Embedded film under shear + is applied at the upper In this case, only shear stress τxz surface (z = tm + ta /2). The lower bonding layer is fixed, i.e. u− = 0, ua (z = −ta /2) = 0. It is assumed that wa = 0 throughout the film segment. A linear variation of shear stress in x is assumed as in Sections 3.2 and 3.3. A linear variation of the upper surface displacement u+ in x is imposed, which is consistent with this assumption. From the displacement continuity at z = ta /2, one can write
u+ =
+ τxz a ta t + u , m G∗ 2
τ+ ua (ta /2) − ua (−ta /2) = xz . ta c55
(36b)
Combining Eqs. (36a) and (36b) and using Eq. (6), we get u0 =
+ ta τxz , 2 c55
φ=
+ τxz . c55
(36c)
Further, Eqs. (15a) and (15b) can be rewritten as + + (−L/2) τxz (L/2) − τxz c11 ta C3 = − , C4 = 0. 2c13 c55 L (36d) Using Eqs. (36c) and (36d) in Eqs. (22), (24) and (25), we get + + (−L/2) (L/2) − τxz ta τxz D1 = c55 L c13 e31 c11 c33 e31 × c11 − + . (37) e33 2c13 e33
3.4. Effect of residual stress
where A = e31 +
and from condition of the shear stress transfer from z = ta /2 to z = −ta /2, one can write
The explicit form of the capacitance is given by Eq. (30). Numerical results using this analytical solution are shown in Section 4.4.
The explicit form of the capacitance is obtained as C=
429
(36a)
In order to include the effect of residual stress in the present model, we consider the expressions for various types of residual stresses as discussed in [8]. In the numerical studies in Section 4.5, we shall consider only three types of residual stresses, namely, the intrinsic, the thermal and the transition stresses. Experimental studies (e.g. [8]) shows that the intrinsic and transition stresses are compressive and thermal residual stresses are tensile in nature. Stresses arising in case of epitaxially grown film, which is called epitaxial stress, is not considered in the present analysis, since it does not contribute to residual stress [19]. Once the total residual stress is estimated by summing up all the individual process-dependent residual stresses, the total equivalent residual strain can be calculated. This total residual strain (εrxx ) can be included in the present model while considering a particular type of constrained boundary. Below we show the formulation for the case of + and σ − applied respectively at in-plane tensile stresses σxx xx the two ends x = ±L/2 of the film as in Section 3.2. The modified forms of Eqs. (33a) including residual strain becomes 1 − σ + + σxx C3 = − 2c11 2c13 xx + u (L/2) − u− (−L/2) (38) × + εrxx . L
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R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437 Table 1 Material properties of the film
Expanded form of Eq. (22) becomes c13 e31 u+ (L/2) − u+ (−L/2) D1 = c11 − e33 L c33 e31 C4 L + c13 + c55 − 2 e33 −
c13 e31 r c33 e31 C3 − ε . e33 e33 xx
(39)
Consequently, the explicit form of the output voltage and the electric displacement can respectively be written from their general form in Eq. (19) and (24) as + u (L/2) − u+ (−L/2) c13 33 r D+ = e + + ε 31 z xx e33 L c33 33 + e33 + (C3 + C4 x), (40) e33 u+ (L/2) − u+ (−L/2) ta c11 $Φ = e31 L + (c13 + c55 )C4 x − D1 .
(41)
∂u0 ta ∂φ + + εrxx . ∂x 2 ∂x
PVDF
PZT
c11 (GPa) c13 (GPa) c33 (GPa) c55 (GPa) Poisson’s ratio (ν) d31 (pC/N) d33 (pC/N) e31 (N/V m) e33 (N/V m) Relative permittivity s33 / 0 at ω = 1 MHz Dissipation factor η Volume resistivity ρ (ohm m)
2.00 0.768 2.00 0.746 0.34 8 15–16 0.02753 0.03615 11 0.018 5 × 1012
69.00 23.664 55.00 26.336 0.31 −175 350 −3.79256 15.1088 1800 0.018 2 × 1010
directly transfered to the sensor film. Also, an additional assumption is made, that is the effective thickness ts of the host structure over which the stress transfer takes place is known. Material properties of the PVDF and PZT film used in the numerical simulations are shown in Table 1. Carbon-epoxy composite (55% fiber volume fraction is used as host structural materials with c11 = 55.1 GPa. The shear modulus of the bonding layer is assumed as G = 1 GPa. The viscoelastic property of this bonding layer is modeled as G∗ = G(1 − iηm ), where the loss factor is ηm = 0.007. The frequency dependent dielectric including the effect of AC loss is given by [20]
Using Eqs. (33b), (38) and (39) in Eqs. (40), (41) and (25), the capacitance of the film segment is given by Eq. (35) with F3 modified as F3 =
Property
(42)
33 = s33 (1 − iη ) −
Numerical results using this analytical results are discussed in Section 4.4.
i , ρr ω
(43)
where s33 is the DC component of the dielectric, η is the AC dielectric loss factor, ρr is the volume resistivity and ω is the AC frequency. In all the numerical simulations, the thickness of the bonding layer tm = 10 m, film width b = 1 mm, film length L = 2 mm and 3 mm respectively for PVDF and PZT, film thickness ta = 10 m and 100 m, respectively for PVDF and PZT. Fig. 2.
4. Numerical results and discussions In this section, we first show the variation in the capacitance and output voltage based on the proposed model (termed as Formulation 1) of the surface-bonded film compared to the results from conventional expression of electrostatics (C = b 33 (L2 − L1 )/ta , termed as Conventional) and another formulation based on simple stress-transfer and strain continuity (termed as Formulation 2). A brief description of Formulation 2 is given in the Appendix. Note that in this simple one-dimensional model, only a constant in-plane tensile stress profile in the host structure is assumed to be
4.1. Surface-bonded film Fig. 3(a) and (b) respectively show the dependence of capacitance of the surface-bonded PVDF film and PZT film on their geometric properties, i.e., film thickness and length. It can be seen that in case of PVDF, the results predicted by dx
Z Host structure
ts
Bonding layer
tm
Film (a)
X
t a /2
s
σxx
σ
xx
s s σ xx + dσ xx
τxz τxz
σ xx + dσxx
(b)
Fig. 2. Schematic diagram of the simplified model of surface-bonded sensor based on stress transfer and strain continuity at the bonding layer.
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
431
Formulation 1 Formulation 2 Conventional
Formulation 1 Formulation 2 Conventional
12.5
60 10
Capacitance (nF)
Capacitance (pF)
50 40 30 20 10
7.5 5 2.5 100
50 40
50 Thickness (µ m)
1.5
20 Thickness (µ m) 10
(a)
75
2 30 0.5
1 Length (mm)
25
0.5
(b)
1.5
1
2.5
2
3
Length (mm)
14
14
12
12
10
10
Voltage (V)
Voltage (V)
Fig. 3. Capacitance of (a) PVDF sensor and (b) PZT sensor surface-bonded on carbon-epoxy composite. Formulation 1: present model (Eq. (30); Formulation 2: direct stress transfer and strain continuity (Eq. (A.10)); Conventional: electrostatics.
8 6 4
8 6 4
2
2
0 6 10
0 6 10 0.5
5
10
10
Stress (Pa)
3
(a)
10
0.25 0.5
0 x/L
0.5
5
10
0.25 4
0.25 0
4
10
Stress (Pa)
3
(b)
10
x/L
0.25 0.5
Fig. 4. Output voltage of PVDF sensor surface-bonded on carbon-epoxy composite obtained from (a) Formulation 1 and (b) Formulation 2.
0
0
- 10
- 50
- 20
- 100
- 30 - 40 - 50
- 150 - 200 - 250
- 60
- 300
- 70 3 10
3
10 4
4
0.5
10
10
10
7
10
0.25 0.5
0.25
5
10
0
6
Stress (Pa)
0.5
10
0.25
5
(a)
affect the constrained electro-mechanical behavior and are modeled more accurately through Formulation 1. Fig. 4(a) and (b) respectively shows the variation in the output voltage of the surface-bonded PVDF sensor on carbon-epoxy composite obtained from the present model and Formulation 2. Similarly, Fig. 5(a) and (b) respectively shows the
Voltage (V)
Voltage (V)
the present model is almost identical to the result from conventional model of electrostatics, but differ from Formulation 2. In case of PZT, the present model significantly differs from both the conventional model as well as from the Formulation 2. This can be attributed to the higher stiffness of the PZT film and its higher thickness range that significantly
0
6
x/L
Stress (Pa)
(b)
10
7
10
0.25 0.5
x/L
Fig. 5. Output voltage of PZT sensor surface-bonded on carbon-epoxy composite obtained from (a) Formulation 1 and (b) Formulation 2.
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Formulation 1 Formulation 2 Conventional
12 10 Capacitance (nF)
Capacitance (pF)
60 50 40 30 20
8 6 4 2
10
0 100
50
80
40
2 30
20 Thickness (µ m) 10
(a)
60
1.5
40
1 Length (mm)
0.5
20
Thickness (µ m)
(a)
0
0
1
0.5
1.5
2.5
2
3
Length (mm)
Fig. 6. Variation in the capacitance of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite due to variation in the film thickness and length.
variation in the output voltage of the surface-bonded PZT film on carbon-epoxy composite obtained from the present model and Formulation 2. For all these results using Formu− (−L/2) = 0 and τ − (L/2) is varied as shown in lation 1, τxz xz the figures. It is clear that the low stiffness of the PVDF film produces similar voltage amplitude predicted by the Formulation 1 and the Formulation 2. However, for PZT which has high stiffness, it can be seen from Fig. 5(a) and (b) that there are differences in the voltages. This difference can be attributed to the fact that in Formulation 2, the admissible in-plane tensile stress is constant throughout the film span, which is not the exact case because of certain amount of horizontal shear deformation across the sensor film thickness. This becomes more significant as the film stiffness increases.
Fig. 6(a) and (b) respectively shows the variation in the capacitance of the PVDF film and the PZT film embedded in carbon-epoxy composite for variation in the thickness and length. The classical prediction is that the capacitance becomes infinite as the distance between electrodes becomes zero [21]. Here also we observe the exponential increase in the absolute value of the capacitance with decreasing film thickness. In order to study the effect of tensile stresses on the + at x = L/2 and σ − at x = −L/2 are varcapacitance, σxx xx ied separately by keeping one of them constant at a time. Fig. 7(a) and (b) respectively shows the percentage variations in capacitance for PVDF film and PZT film embedded in carbon-epoxy composite with respect to the conventional estimate. These results indicate that for different combinations of tensile stresses applied at x = ±L/2, the maximum percentage variation in capacitance, compared to the conventional, is nearly 1% for PVDF which has low stiffness and nearly +65 to −40% for PZT which has higher stiffness. Also, it can be seen that when the applied stresses at the ends are nearly equal, the percentage variations in capacitance remain nearly constant for all stress levels.
4.2. Embedded film subjected to tensile stress
1.25
% Variation in Capacitance
% Variation in Capacitance
+, σ− Here, first the effects of in-plane tensile stresses σxx xx and the film length on the capacitance and output voltage are + is studied studied. The effect of transverse normal stress σzz next. In all these cases, the linear spatial variation of the applied tractions are considered as discussed in Section 3.2.
1 0.75 0.5 0.25 0 6 5 4 Log10(σ-xx)
(a)
5
3 2 1
1
2
3
4 Log10(σ+xx)
6
75 60 45 30 15 0 15 30 45 7 6
7
5 4
(b)
Log10(σ-xx)
5
3 2 1
1
2
3
6
7
8
4 + Log10(σxx)
+ and σ − on capacitance of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite. Fig. 7. Effect of tensile stresses σxx xx
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
100 0 - 100 - 200 - 300 - 400 - 500 - 600 - 700 - 800 3 - 10
Voltage (V)
Voltage (V)
800 700 600 500 400 300 200 100 0 100 6 10 0.5
5
10
10
Stress (Pa)
3
(a)
10
0.25
0.25 0
5
Stress (Pa) - 10
x/L
0.5
0.5
4
- 10
0.25 0
4
433
0.25
x/L
0.5
10 6
(b)
600
600
500
500
400
400 Voltage (V)
Voltage (V)
+ at x = +L/2 and (b) σ − at x = −L/2. Fig. 8. Output voltage of PVDF sensor embedded in carbon-epoxy composite under varying tensile stress (a) σxx xx
300 200 100 0
300 200 100 0 - 100
100
- 200 7 - 10
200 7 10 6
6
0.5
10
Stress (Pa)
4
10
3
(a)
10
0.25 0.5
0.5
- 10
0.25
5
10
0.25
5
- 10
0 x/L
Stress (Pa)
0
4
- 10
(b)
10
3
0.25
x/L
0.5
+ at x = +L/2 and (b) σ − at x = −L/2. Fig. 9. Output voltage of PZT sensor embedded in carbon-epoxy composite under varying tensile stress (a) σxx xx
ficient contribute to certain amount of bias field along the film span causing loss of symmetry for the same stress field. 4.3. Embedded film subjected to transverse normal stress In order to study the effect of transverse normal stress, + at the upper surface z = t + t /2 we apply constant σzz m a
0
0
- 50
- 10 - 20
- 100
Voltage (V)
Voltage (V)
Corresponding output voltages for varying stresses at the ends separately can be seen in Fig. 8 for PVDF film and in Fig. 9 for PZT film. It can be seen that for PVDF, higher voltages are developed towards the end segments which are subjected to higher stresses. The response is in direct correlation with the strain induced due to stress field. On the other hand, for PZT, the high stiffness and the polarization coef-
- 150 - 200
- 30 - 40 - 50
- 250
- 60
- 300 3 10
- 70 3 10
(a)
6
10
0.25 0.5
0.25
5
10
0
10
0.5
10
0.25 5
Stress (Pa)
4
0.5
4
10
0
6
x/L
Stress (Pa)
(b)
10
7
10
0.25
x/L
0.5
+ at Fig. 10. Output voltage of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite under varying transverse normal stress σzz z = tm + ta /2.
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
7
7
6
6
5
5
Voltage (V)
Voltage (V)
434
4 3 2
4 3 2
1
1
0 6 10
0 7 10 6
0.5
5
10
4
Stress (Pa)
10
3
(a)
10
0.25 0.5
0.5
10
0.25
0.25
5
10
0 x/L
0
4
Stress (Pa)
10
3
(b)
10
0.25
x/L
0.5
+ at z = +t /2. Fig. 11. Output voltage of (a) PVDF sensor and (b) PZT sensor embedded in carbon-epoxy composite under shear stress τxz a
while keeping the lower surface restrained (w− = 0). In this case, the expression of capacitance is same as in Eq. (35) + /e . All the other boundary tractions are with F1 = σzz 33 33 zero in the corresponding expressions given in Section 3.2. Fig. 10(a) and (b) respectively shows the output voltage of the PVDF film and the PZT film embedded in carbon-epoxy composite. Since, the applied traction is constant over the film length, the voltage is also constant. However, with the applied stress typically in the order of MPa, higher magnitude of voltage in 10µm PVDF compared to 100 m PZT can be seen.
ical work conjugate are decoupled from the electrical work conjugate i.e. e51 = 0. As it can be noted from Section 3.3 that the above simplification has made the present approach adequate to deal with arbitrary shear traction distribution + and τ − . Following further simplification of the analytiτxz xz cal solution, the effect of linear shear traction on the voltage is studied. Although, the mechanical and electrical work conjugates are decoupled, the boundary constraints on the embedded film indeed produce a voltage, which is shown in Fig. 11(a) and (b) respectively for PVDF and PZT films. Here, the applied shear stress is varying linearly from x = −L/2 to x = +L/2. This shear stress distribution results in a constant stress gradient along the film-substrate interface. From Eqs. (24), (36d) and (37), it is clear that the output voltage is a function of shear stress gradient. However, Fig. 11 shows that due to decoupled shear and electric field, the magnitude of voltage is quite less as compared to that in cases of equivalent order of applied tensile stress (Figs. 8 and 9) and applied transverse normal stress (Fig. 10).
4.4. Embedded film subjected to shear stress In Section 3.3, a solution has been obtained, where the embedded film is subjected to pure shear stress at the upper and lower interfaces between the bonding layer and the host structure. This model is based on the constitutive relation, where unlike anisotropic piezoelectric material, the mechan-
20 18 16
14
Capacitance (nF)
12 Voltage (V)
10 8 6 4 2 0 2 7 10
With residual stress Conventional Without residual stress
14 12 10 8 6 4
6
10
5
10 Applied tensile (a) stress (Pa)
4
10
3
10
0.5
0.25
0.25
0 x/L
2
0.5
0 3
(b)
3.5
4
4.5
5
5.5
6
6.5
7
+
Applied tensile stress (log[σxx])
Fig. 12. (a) Output voltage of 1 mum PZT film under varying applied tensile stress along with compressive residual stress (shown by thick solid lines) and without the residual stress (shown by thin solid lines). (b) Effect of residual stress on capacitance of 1 m PZT film at various mechanical tensile stress levels.
R. Ali et al. / Sensors and Actuators A 116 (2004) 424–437
4.5. Effect of residual stress Here we show the comparative response of a 1 m PZT film deposited on Pt/Ti/Si(0 0 1), with and without the residual stress. The explicit form of the output voltage and the capacitance have been obtained in Section 3.4. The values of residual stress components are calculated on the basis of formulae given in [8]. As discussed in Section 3.4, the values of intrinsic, thermal and transition stress are estimated as 117.2, 217 and 110.2 MPa, respectively. Therefore the total residual stress is sum of these three values, which is found to be 10.2 MPa compressive. In the present calcula+ and σ − are applied tions, the mechanical tensile stress σxx xx at the two ends x = ±L/2 are made equal. The compressive residual stress estimated above is applied through the equivalent residual strain using Eq. (41). Fig. 12(a) shows the the variation in the voltage at various locations of the film under varying tensile stress. The constant tensile stress over the film length is increased from 1 kPa to 10 MPa. It can be observed from Fig. 12(a) that the voltage in presence of the estimated compressive residual stress is approximately 4 V, which is higher than the voltage without the residual stress. This can be attributed to some amount of residual charge which is already stored on the film surface due to induced residual strain. It is not only the voltage that is affected by the residual stress, but also the capacitance, which is clear from Fig. 12(b). From this figure, it can be observed that the capacitance including the effect of residual stress is nearly 12 nF lesser than the capacitance without the residual stress. However, when the applied tensile stress is in the order of 1–10 MPa, the effect of compressive residual stress is nullified and the capacitance drops showing a very small amount of transduction caused by mechanical deformation. In contrast, the conventional estimate is independent of the effect of any applied or residual stress. According to the definition of the capacitive sensor, it is the capacity of the sensor material to accumulate the electrical charge. The present formulation and the above numerical results clearly show that the equivalent capacitance of a piezoelectric film is actually dependent on the constrained boundary, applied mechanical load and also the residual stress.
5. Conclusions A new analytical model has been reported to analyze the performance of piezoelectric thin film sensors, surface-bonded or embedded in composite material system. As an improvement over the existing models, the proposed model includes several non-classical effects, such as viscoelastic property of the bonding layer, AC dielectric loss and frequency-dependent effect of the resistivity of the film materials, and more importantly, the effect of constrained boundary. Governing equations are solved exactly by considering first-order shear deformation. General expression for the electric charge displacements and potential
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difference are obtained. The boundary constraints can be calculated a-priori using appropriate analytical solution, or in more detail, by boundary layer stress and displacement extraction from finite element analysis of the host structure. The analytical solution shows significant complexity of the functional dependence of various parameters, such as, layer geometry, combination of material properties and stress. Four cases are solved and studied numerically, which are, (1) when the film is on the surface of the host structure, (2) when the embedded film is subjected to in-plane tensile stress, (3) when the embedded film is subjected to transverse normal stress and (4) when the embedded film is subjected to pure shear. In all these cases, the performance of PVDF film and PZT film is studied for a wide range of applied stress. In case of surface-bonded film, the results are compared with existing models. For the cases of embedded film, the variations in the voltage due to change in the stresses are found to be significant, which are otherwise difficult to simulate using exiting model. The results indicate significant trade-off that may be necessary in design optimization of piezoelectric thin film device for various applications. Also, a clear numerical illustration is made to show the effect of residual stress on the capacitance and voltage. This is an important effect to consider, which is a common issue in most of the MEMS and thin-film fabrication and packaging processes. The proposed model can be useful while dealing with complex mechanical loading. The analytical solution can also be useful in optimizing the capacitive performance. Additional complexities, such as the nano-scale effects, may be possible to incorporated in the present model.
Appendix A. A simple model based on stress transfer and strain continuity A piezoelectric film bonded to the surface of the host structure is considered. A constant in-plane tensile stress profile in the host layer is assumed to be directly transfered to the film along with the condition of strain continuity at the film-substrate interface. The free-body diagram is shown in Fig. 2. The stress transfer mechanism is based on shear-lag model [20]. The superscript s indicates the variables in the host/ substrate structure. The superscript a indicates the variables in the film. As shown in Fig. 2, ts , tm and ta /2 are respectively the thicknesses of the host structure (an equivalent thickness for effective stress transfer), the thickness of the bonding layer and the thickness of the film. Considering a span-wise segment of infinitesimal length dx of the film along with the bonding layer, the equilibrium equations can be written as dσxx ta b + 2b dxτxz = 0,
s dσxx ts b − b dxτxz = 0.
(A.1)
where τxz is the shear stress transferred from host structure to the film through the bonding layer. From the shear-lag
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model, we have us − u a τxz = G∗ . tm
(A.2)
Differentiating Eq. (A.1) thrice with respect to x and subs = cs εs , yields after simplistituting σxx = c11 εxx and σxx 11 xx fication, 2 s d4 εsxx 2 d εxx − Γ = 0, 0 dx4 dx2
where Γ02
G∗ = s c11 ts tm
2 d4 εxx 2 d εxx − Γ = 0, 0 dx4 dx2
s t +c t 2c11 s 11 a c11 ta
(A.3)
.
(A.4)
The general solution for strains in Eq. (A.3) is given by εxx = C1 + C2 x + C3 sinh (Γ0 x) + C4 cosh(Γ0 x), εsxx = C1 + C2 x −
c11 ta s t (C3 sinh (Γ0 x) + C4 cosh(Γ0 x)) . 2c11 s (A.5)
Boundary conditions are now applied separately on the host structure and on the film, which are σs L εsxx (x) = sxx , σxx ± = 0. (A.6) c11 2 The unknown constants are then expressed as s σxx s (1 + (c t /2cs t )) , c11 11 a 11 s C1 C4 = − . cosh(Γ0 (L/2))
C1 =
C2 = 0,
C3 = 0, (A.7)
For pure sensory transduction, i.e., assuming all the stresses are transformed into electric potential and dissipated through the circuit, σxx = c11 εxx − e31
$Φ = 0, ta
Dz = e31 εxx + 33
$Φ . ta (A.8)
Surface charge accumulated over the film segment of width b and span (x = −L/2, +L/2) is given by 2 e31 bc11 σxx Q= + 33 s s t )) e31 c11 (1 + (c11 ta /2c11 c11 s +L/2 cosh(Γ0 x) 1− × (A.9) L dx. cosh Γ0 2 −L/2 Equating the deformation-induced electric power, the equivalent capacitance of the film segment L is obtained as 2 e31 b 4 sinh (Γ0 (L/2)) C= L− + 33 ta c11 cosh(Γ0 (L/2))Γ0 2 sinh (Γ0 L) 1 + + 2L . (A.10) Γ0 4cosh2 (Γ0 (L/2))
In absence of any piezoelectric effect, the conventional form of electrostatic capacitance can be clearly seen from Eq. (A.10). References [1] S.D. Senturia, Microsystems Design, Kluwer Academic Publishers, 2000. [2] M. Yocum, H. Abramovich, A. Grunwald, S. Mall, Fully reversed electromechanical fatigue behavior of composite laminate with embedded piezoelectric actuator/sensor, Smart Mater. Struct. 12 (2003) 1–9. [3] C.V.S. Sastry, D. Roy Mahapatra, S. Gopalakrishnan, T.S. Ramamurthy, Effect of actuation phase on the SIF in delaminated composite with embedded piezoelectric layers, Smart Mater. Struct. 12 (2003) 602–611. [4] C.V.S. Sastry, D. Roy Mahapatra, S. Gopalakrishnan, T.S. Ramamurthy, Distributed sensing of static and dynamic fracture in self-sensing piezoelectric composite: finite element simulation, Intell. Mater. Syst. Struct. 15 (2004) 339–354. [5] D.F. Bahr, J.S. Robach, J.S. Wright, L.F. Francis, W.W. Gerberich, Mechanical deformation of PZT thin films for MEMS applications, Mater. Sci. Eng. A 259 (1999) 126–131. [6] D.F. Diao, K. Kato, K. Hokkirigawa, Fracture mechanisms of ceramic coatings in indentation, ASME J. Tribol. 116 (1994) 860–869. [7] A. Steegen, K. Maex, Silicide-induced stress in Si: origin and consequences for MOS technologies, Mater. Sci. Eng. R38 (2002) 1–53. [8] Y.C. Zhou, Z.Y. Yang, X.J. Zheng, Residual Stress in PZT thin films prepared by pulsed laser deposition, Surf. Coatings Technol. 162 (2003) 202–211. [9] T.Y. Zhang, L.Q. Chen, R. Fu, Measurements of residual stresses in thin films deposited on silicon wafers by indentation fracture, Acta Mater. 47 (14) (1999) 3869–3878. [10] J.W. Hutchinson, Z. Suo, Mixed mode cracking in layered materials, Advances in Applied Mechanics, vol. 29, 1992, Academic Press, New York, pp. 63–191. [11] A.G. Evans, J.W. Hutchinson, The thermomechanical integrity of thin films and multilayers, Acta Metall. Mater. 43 (7) (1995) 2507–2530. [12] W.D. Nix, Mechanical properties of thin films, Metall. Trans. 20 (1988) 2217–2245. [13] E. Sushir, Interfacial stresses in bimetal thermostats, ASME J. Appl. Mech. 56 (1989) 595–600. [14] E. Sushir, Analysis of interfacial thermal stresses in a trimaterial assembly, J. Appl. Phys. 89 (7) (2001) 3685–3694. [15] T.-C. Chen, H.-C. Wu, C.-L. Lin, Longitudinal anisotropic stress and deformation in multilayered film heterostructures due to lattice misfit, J. Cryst. Growth 249 (2003) 44–58. [16] G. Schrag, G. Zelder, H. Kapels, G. Wachutka, Numerical and experimental analysis of distributed electromechanical parasitics in the calibration of a fully BiCMOS-integrated capacitive pressure sensor, Sens. Actuators A (physical) 76 (1999) 19–25. [17] N. Mokni, F. Sidoroff, A. Danescu, A one-dimensional viscoelastic model for lateral relaxation in thin films, Comput. Mater. Sci. 26 (2003) 56–60. [18] T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, New York, 1990. [19] C.M. Foster, Z. Li, M. Buckett, D. Miller, P.M. Baldo, L.E. Rehn, D.R. Bai, D. Guo, H. You, K.L. Merkle, Substrate effects on the structure of epitaxial PbTiO3 thin films prepared on MgO, LaAlO3 and SrTiO3 by metalorganic chemical vapor deposition, J. Appl. Phys. 78 (4) (1995) 2607–2622. [20] A. Pizzochero, Residual actuation and stiffness properties of piezoelectric composites: theory and experiment, M.S. Thesis, MIT, 1998. [21] C. Trimarco, The electric capacitance of a rigid dielectric structure, Int. J. Solids Struct. 29 (13) (1992) 1647–1655.
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Biographies R. Ali is currently working at the aerospace engineering department, Indian Institute of Science (IISc), Bangalore, as research assistant in the field of modeling of thin films for MEMS and SHM applications. Prior to this, he worked at Advanced Composites Division of National Aerospace Laboratories, Bangalore, India, as Research Assistant, in the field of finite element modeling of shape memory alloy wires embedded in composites for actuators in aerospace applications. He received his ME degree in Aerospace Engg. from IISc, in January 2002. His research interests are in smart structures, composites, non-destructive testing and structural health monitoring (SHM) of aerospace and industrial structures. D. Roy Mahapatra received his BE degree in Civil Engineering from Jadavpur University, Calcutta, India, in 1998 and received his PhD in Aerospace Engg. from Indian Institute of Science (IISc), Bangalore in 2004. Currently, he is working as a postdoctoral fellow in the department of Aerospace Engg., IISc. His current areas of research are on the
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development of finite element techniques for analysis of vibration and wave propagation in composite structures, design of functional composite for vibration sensing and control applications, design of micro and nano systems, interaction of wave with materials, shock wave and hypersonics. S. Gopalakrishnan received his BE degree in Civil Engineering from the Bangalore University, India in 1984 and MTech degree in Engineering Mechanics from the Indian Institute of Technology (IIT), Chennai in 1987. He received his PhD. from the School of Aeronautics and Astronautics, Purdue University, West Lafeyette, USA in 1992. Later, he was a post doctoral fellow in the School of Mechanical Engineering at Georgia Tech., Atlanta, USA, from January 1993 to July 1994. He then became an Assistant Professor in the Department of Civil Engineering, IIT, Chennai. He joined the Department of Aerospace Engineering at the Indian Institute of Science (IISc), Bangalore as an Assistant Professor in November 1997. He became an associate professor in the same department in 2003 and continuing in this position. His main areas of interest are structural dynamic, wave propagation, smart structures and FEA.