Numerical simulations of piezoelectric effects

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1 Numerical simulations of piezoelectric effects Raffaele Ardito, Alberto Corigliano, Giacomo Gafforelli Department of Civil and Environmental Engineering, Politecnico di ilano, ilano, Italy Keywords: ulti-physics simulations, Piezoelectric actuators, Energy harvesters. SUARY. This paper is focused on the numerical simulation of piezoelectric effects, in view of their application in ES harvesters and actuators. In order to obtain a fast and accurate prediction of dynamic behavior, a simple -DOF model has been introduced for thin piezoelectric beams. The validation of the model has been carried out by comparison with full D simulations via commercial codes.. INTRODUCTION The use of piezoelectric materials is steadily increasing in the field of smart structures and of icro-electro-echanical Systems (ES). In particular, the coupling between electrical field and mechanical behavior can be used in the design of ES energy harvesters (which eploit the so-called direct effect []) and actuators (based on the indirect effect []). Piezoelectric ES have been proven to be an attractive technology for harvesting small magnitudes of energy from ambient vibrations. This technology promises to eliminate the need for batteries or comple wiring in microsensors/microsystems, moving closer towards battery-less, autonomous sensors systems and networks which recovered on-site the energy they need to fulfill their tasks. New developments in sensors and complementary metal oide semiconductor (COS) circuitry technology have considerably reduced the power consumption of electronics which can be now powered by harvesters able to robustly generate about 00 µw of continuous power from ambient vibrations []. In addition, the device should be small enough to be integrated with the electronics and the cost should be sufficiently low for mass scale deployment. For ES-scale (smaller than 0.5 cm ) energy harvesters, piezoelectric transduction is the most appropriate scenario since standard ES thin-film processes are available for many piezoelectric materials assuring high efficiency, high energy density and scalability. Operating frequency, frequency bandwidth, ecitation level, power density and size are the key design function requirements. Cantilever laminated beams with thin films of lead zirconate titanate Pb(Zr,Ti)O (PZT) have been widely used as linear resonating harvesters achieving high power generation both through d- and d-mode []. On the other hand, the piezoelectric effect could be used for actuation purposes in several applications (e.g. fluid pumping at the microscale, [4]). The multi-physics simulation of piezoelectric effect can be obtained by considering that the structural members are represented by a laminate composite with piezoelectric and silicon layers, the piezoelectric material is then attached to an eternal load resistance which reproduces the circuitry employed for the power management. The mechanical response can be obtained by means of a number of theories, eamined in [5]. In this paper the sectional behavior of the beam is studied through the Classical Lamination Theory (CLT) specifically modified to introduce the piezoelectric coupling. A lumped parameters model is built through Rayleigh-Ritz ethod. The coupled equations of linear dynamic piezoelectricity are solved in the frequency domain by means

2 of Harmonic Balance ethod (HB) and numerically solved in the time domain by means of Hughes α method [6]. For the sake of comparison, simulations on a mesoscale device have been carried out also by means of fully D model and commercial codes, properly modified in order to introduce the effect of the electrical circuit. Even if the validation is performed at the mesoscale, the validated -DOF model can be used to characterize ES scale device.. PIEZOELECTRIC HARVESTER ODEL Starting from kinematic assumptions and piezoelectric constitutive law a -DOF structural model of piezoelectric energy conversion is built. Two physical domains are involved: the piezoelectric, which involves the electro-mechanical coupling, and the electrical domain... Geometry The piezolaminated cantilever beam is presented in Figure where L is the length and h the total thickness. The -coordinate originates in the neutral ais and is directed downwards while -coordinate lies along the beam ais. A PZT layer is placed on top of the beam substrate and is activated in d-mode when the beam deflects. To implement d-mode electrodes span both surfaces of PZT thin film, and polarization is directed opposite to -ais. In such way when the PZT layer is stretched along -ais an electric field arises parallel to the -ais and the generated charge is collected by the electrodes. Figure : Cantilever piezoelectric harvester The bottom electrode is grounded while the other one is attached to an eternal load resistance (R) which represents an eternal circuit used for the management of the power generated by PZT... Sectional behavior: odified Classical Lamination Theory (CLT) The beam final stack is not homogeneous since different deposited layers are employed. In addition to those above mentioned, a barrier layer is required to avoid diffusion of PZT in the structural layer, a seed layer is needed to control PZT growth and passivation layers are necessary to assure the good electrical behavior of the system. The mechanical response of the layered beam can be obtained by means of a number of theories, eamined in [5]. Very thin beams are usually employed in ES devices, thus Bernoulli hypothesis and CLT is adopted herein, with the important difference of introducing the piezoelectric coupling in PZT layer. The rotation of the cross-section is the derivative of vertical displacement (w ) and the horizontal displacement (s ) and strain (S ) results:

3 w s (, ) = ( ) w ( ) (, ) = S () The electric potential is constant over the electrodes: the potential value is assigned on the bottom electrode (grounded electrode) and it is free to change on the upper (v). According to piezoelectric constitutive law, the electric field is proportional to strain which is linear across the piezoelectric layer thickness, thus the electric potential (φ ( )) should be considered quadratic across the thickness of piezoelectric. However, as long as the piezoelectric layer is thin, a linear potential can be adopted and consequently a constant electric field E is obtained: * φ ( ) = v E tp φ v = = t P () * where t p is the piezoelectric layer thickness and ( ) piezoelectric layer (Figure ). is the vertical upward coordinate in the Figure : Polarization and electric potential between electrodes Piezoelectric constitutive law is adopted herein to describe strain-stress relation, the fullycoupled law for d-mode reads as follows: S T = c S e E D = e S + ε E () where T is the stress component parallel to -ais, S is the strain tensor component, D the S electric displacement component, E the electric field component and c, e and ε the elastic, piezoelectric and dielectric constant, respectively. Eqs. () are known as the e-form of piezoelectric constitutive relations. The adopted notation is customary in the theory of piezoelectricity []. By considering the stratification of Figure, where,k is the coordinate of the k-layer and n l the total number of layers, and by integrating on the thickness, one obtains the generalized internal stiffness ( C ηχ, C χχ ) usually defined for the theory of laminates. In such a case the new constitutive law contains, in the integrated constitutive equations, the generalized piezoelectric coupling coefficient ( ). oreover, the same procedure is adopted on the electrical part of the C χ v constitutive law, obtaining the generalized electrical capacity ( C ) [4]. vv

4 Cηχ = c d = c C = c d = c, k, k k k k k (, k, k ) χχ (, k, k ) k = k = k = k, k =, k k k, S, k S, k e e ε C = d = C = d, k k (,, ) = (, k, k ) k k vv k = t k, k P = tp k = t, k P k = tp ε (4) The translational and rotational mass of the beam is based on ρ k, the density of the k-th layer: m = d = m = d =, k, k k k k k ρ ρ (,, ) ρ ρ (,, ) (5) k k k k k = k = k = k, k =, k Figure : Piezolaminated beam stratification.. Lumped model: Rayleigh-Ritz method Rayleigh-Ritz method is adopted to describe beam deflection along the beam on the basis of the tip displacement w(t), whilst the electric potential v is kept constant over the beam length: and: The principle of virtual power writes: ( ) ( ) ψ ( ) w t = w t (6), w ( δ, δ ) ( δ ) = ( δ, δ ) + ( δ &, δ & ) P w& v& P w& P s s P S E (7) et diss inertia int, k, k dw& k ds& k ds& Pinertia = mtip δ w& + s d da s d da dt ρ δ & + A k dt ρ δ & = k, k A = dt, k P = f δ w& da + f δ w& + q δ v& P = c δ w& et DIS TIP diss A, k k k k k Pint = ( T δ S& + D δ E& ) d da A k =, k ( ) (8)

5 where c is the mechanical damping coefficient, f DIS is the uniformly distributed force on the beam surface, f is the eternal concentrated force at the free edge of the beam and q is the TIP total electric charge collected by the upper electrode. By employing eq. (7) and (8) and using the hypotheses herein adopted (sections. and.) the dynamic equilibrium equations of the coupled system results: m w&& + c w& + k w Θ v = f m && y E L et k v + Θ w = q (9) Eq. (9) describes the dynamical behavior of the linear harvester. The coefficients are computed by integrating the shape functions and the generalized constitutive coefficients on the area of the beam: ψ ψ m = m + b m d + b m d f = f + b f d L L L w w TIP ψ wψ w TIP DIS ψ w ψ ψ ψ k b C d k b C d b C d L L L w w w L = χχ E = vv Θ = (0) where m is the total mass; kl is the linear elastic stiffness; while Θ is the linear coupling coefficient. k E is the internal capacitance of PZT The electric charge collected by the electrodes is managed by an eternal circuitry which provides the power supply for the self powered electronic device. Different schemes of circuitries are investigated in [7]. The harvester provides AC voltage and simplest solution is the coupling with an eternal load resistance: & () q = R v Eq. (9) and () are usually studied separately by the commercial codes because they belong to two different physical domains which interact through electric potential and electric charge. The solution technique developed for commercial codes is eplained in section. In case of the -DOF code developed herein, the two physics are merged in a unique package and solved simultaneously..4. Solution in the frequency domain: Harmonic Balance ethod (HB) The oscillator frequency response to harmonic ecitations is studied through HB [8] which provides the eact solution since the system is linear. Harmonic balance mimics the spectrum F sin ωt + φ ) is a analyzer in simply assuming that the response to a sinusoidal ecitation ( ( F ) sinusoid at the same frequency. A trial couple of solution ( w = W sin ( ωt), v V sin ( ωt φ ) = + ) is substituted in the equations, the coefficients of same harmonics are equated and the Frequency Response Function (FRF) is computed: v

6 κ χ κ Ω Y ( v χ Ω ) ( v E = Ω i ζ + Ω Ω + Ω E Ω E ) + Ω () n where Y = W / W0 is the dimensioess amplitude, W0 = F / kl is the static displacement, Ω = ω / ω is the dimensioess ecitation frequency, Ω = / Rk ω is the dimensioess cutoff frequency of the circuit ( E E E n Rk is the time constant of the circuit) and ( ) / κ = Θ / k k the E L effective piezoelectric noinear coupling coefficient. κ is a global measure of the degree of coupling which includes d-mode piezoelectric coupling coefficient e / ( c ) / κ = ε and geometrical aspects. In case of a pure piezoelectric system κ would eactly coincide with κ. Eq. () shows that both the equivalent stiffness and damping ratio coefficient depends on piezoelectric coupling and on the eternal ecitation frequency and load resistance: keq = k L + κ Ω κ χ v ΩE ζ = ζ + ζ = ζ + () eq E ( Ω + Ω E ) ( Ω + ΩE ) The power generation is the power dissipated by the equivalent electrical dashpot and results: κ ΩEΩ ( Ω + ΩE ) F ωn ( E )( ) κ ( ζ ( ) κ χ ) P = ce w& = m Ω + Ω Ω + + Ω + Ω + Ω + Ω { E v E Ω }.5. Numerical solution by means of Hughes α-method In the previous section the frequency response of the oscillator is analyzed. However, the eternal ecitation is usually random. Hughes α method [6] has been chosen to solve eq. (9) for time dependent ecitation. Eq. (9) is rearranged in a matri formulation in the variable [ w v] T = : && + C& + K = f (5) (4) where: m 0 c 0 kl Θ f m && yet = 0 0 C = = = k K Θ E 0 R f 0 (6). COERCIAL CODES To validate the model eplained in section, two commercial codes, COSOL and ABAQUS, are employed. Both codes allow using piezoelectric constitutive equations in the

7 classical structural mechanic environment. However, in both codes oy D models can be built, this is an enormous limitation since piezoelectric elements are usually employed in laminated beams or plates. This results in a big numerical effort since a high number of elements must be employed to correctly reproduce the behavior of high aspect ratio elements. oreover, while in COSOL the solver of lumped electrical circuitry is already available, ABAQUS is not provided with this tool. For this reason the connection with the electric circuit has been built ad hoc through an eternal subroutine which tries to replicate the method implemented in COSOL. 4. ANALYSIS To validate the model developed in section, three types of analysis are developed: (i) static, (ii) forced oscillations and (iii) free oscillations. A mesoscale laminated (steel/pzt) has been modeled (length 80 mm, width 0 mm and thickness of both layers 0.5 mm). The material properties are listed in Table I: g / cm [ ] ρ c GPa v[ ] e [ N mv ] e [ N mv ] [ ] / / ε rel Steel PZT Static Analysis Table I: aterials properties The beam is statically deformed by a concentrated force at the free edge and the deformation is kept constant. Two load resistances are considered, the open circuit condition ( R = ) and R = 0 Ω.Figure 4 and Figure 5 show amplitude of the free edge of the beam and electric potential on the upper electrode. In open circuit condition (Figure 4) the piezoelectric material is not coupled with the eternal load resistance, no charge is moving between electrodes and the voltage remains constant. When decreasing R the coupling with the eternal circuit becomes more and more important. The charge is now free to move between the electrodes through the resistance and the voltage decreases till the electrical equilibrium is reached. In both cases the - DOF laminated model underestimates the voltage. This is due to three-dimensional effects near the clamped edge which are neglected in the modeling. Indeed, the commercial codes manage the whole piezoelectric coupling matri while the -DOF model takes into account oy the component. Figure 4: Static response for in open circuit condition

8 4.. Free oscillations Figure 5: Static response with R=0Ω The beam is statically deformed by a concentrated force at the free edge and then suddey released. Two load resistances are considered, the open circuit condition ( R = ) and R = Ω. The mechanical quality factor is Q = 85. Figure 7 shows that the voltage is almost zero after the static step both for the -DOF model and the commercial codes. The different variation is amplitude in the free dynamics is again due to D effects near the clamping. Good agreement between -DOF and commercial codes is obtained for high mechanical damping ( Q = 8.5 ), too (Figure 8). Figure 6: Free oscillations in open circuit condition Figure 7: Free oscillations with R=Ω and Q=85

9 Figure 8: Free oscillations with R=Ω and Q= Forced oscillations The beam is ecited by a sinusoidal concentrated force at the free edge. The resonance frequency of the beam is ω n = 446Hz. The response of the -DOF model and the codes is studied for different ecitation frequencies ( ω = 00 Hz; 40 Hz; 500 Hz; 800Hz ). Figure 9: Voltage response for forced oscillation with and R=0Ω for ω=00hz and ω=40hz Figure 0: Voltage response for forced oscillation with and R=0Ω for ω=500hz and ω=800hz The -DOF model reproduces correctly the response of the harvester; differences with the commercial codes arise near the resonance frequency of the beam.

10 5. CONCLUSIONS A -DOF simple model has been built to perform multi-physics simulation for ES piezoelectric cantilever harvesters. The piezolaminated beam has been studied through the Classical Lamination Theory (CLT) specifically modified to introduce the piezoelectric coupling and a lumped parameters model has been built through Rayleigh-Ritz ethod. The coupled equations of linear dynamic piezoelectricity have been solved in the frequency domain by means of Harmonic Balance ethod (HB) and have been numerically solved in the time domain by means of Hughes α method. The aforementioned model has been validated by comparison with fully D multi-physics simulation performed by commercial codes (ABAQUS and COSOL). An eternal subroutine has been added to ABAQUS to simulate the coupling with an eternal load resistance. The comparison with the commercial codes shows that the -DOF model is accurate enough to reproduce qualitatively and quantitatively the response of the harvester gaining in terms of model dimensions and computational time. The validate model can be used to perform simulations on ES scale piezoelectric energy harvesters. 6. ACKNOWLEDGEENTS The contribution of the former master students Alessandro orbio and Andrea Pepe is gratefully acknowledged. This research has been carried out with the financial support of the Italian inistry of Education through the grant Prin09, Project n 009XWLFKW: ulti-scale modeling of materials and structures. References [] IEEE Standard on Piezoelectricity, 987: The Institute of Electrical and Electronics Engineers. [] Kim, S.G., Priya, S., Kanno, I. "Piezoelectric ES for energy harvesting." RS Bulletin 7 (), 0: [] Jeon, Y.B., Sood, R., Jeong, J.H., Kim, S.G. "ES power generator with transverse mode thin film PZT." Sensors and Actuators, A: Physical, 005: ( SPEC. ISS.), pp. 6-. [4] Ardito, R., Bertarelli, E., Corigliano, A., Gafforelli, G. "On the application of piezolaminated composites to diaphragm micropumps." Composite Structures 99, 0: -40. [5] Ballhause, D., D'Ottavio,., Kröplin, B., Carrera, E. "A unified formulation to assess multilayered theories for piezoelectric plates." Computers and Structures 8 (5-6), 005: 7-5. [6] Hughes, T.J.R. The finite element method. Englewood Cliffs, NJ: Prentice Hall, 987. [7] Guyomar, D., Badel, A., Lefeuvre, E., Richard, C. "Towards energy harvesting using active materials and conversion improvement by noinear processing." IEEE transactions on ultrasonics, ferroelectrics, and frequency control 5 (4), 005 [8] Worden, K., Tomlinson, G.R. Noinearity in structural dynamics: Detection, Identification and odelling. London, UK: Institute of Physics Publishing, 000.