Micromechanical modeling and simulation of piezoceramic materials

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1 Micromechanical modeling and simulation of piezoceramic materials B. Delibas 1, A. Arockia Rajan 1 & W. Seemann 2 1 Workgroup for Machine Dynamics, Technical University of Kaiserslautern, Germany 2 Institut für Technische Mechanik, University of Karlsruhe, Germany Abstract Piezoceramics are nowadays very well known smart materials, which are commonly used in many engineering applications. Although they are so excellent in usage, they are limited in high performance applications. Loading of piezoceramic materials above a certain level leads to a nonlinear behavior. One reason for this behavior is the ferroelectric and ferroelastic domain switching. In this work, certain piezoelectric materials having tetragonal perovskite type microstructure characteristics are simulated using a micromechanical approach in which all calculations, both the linear constitutive model and the nonlinear switching model are done in each and every grain of the material. Uni-axial loading is applied in the simulations. Macroscopic values for polarization and strain of the material are determined by transforming from local to global coordinates and averaging those that are calculated for the individual grains. The simulated electric displacement hysteresis curve taking into account the various different criteria such as energy differences for the type of switching, different random generators, and changing material parameters for ferroelectric and piezoelectric materials are shown. Finally, simulation results are compared with data that were measured in previous experimental works and which are given in literature. Keywords: piezoelectricity, ferroelectricity, micromechanical modeling, domain switching. 1 Introduction Ferroelectric and piezoelectric materials are used widely as actuators and sensors. The most common applications are vibration control, precision

2 78 High Performance Structures and Materials II positioning, cutting and microelectromechanical systems (MEMS). Piezoceramic materials are grouped into two kinds of materials, the soft and hard piezoceramics. In nature, piezoceramic materials mostly exhibit tetragonal and rhombohedral phases. BaTiO 3 and PLZT are some example of piezoceramic materials which have a perovskite type tetragonal microstructure (Fig.1). The basics of piezoceramic and ferroelectric materials are explained by Jaffe et al. [1]. Piezoceramic materials show a nonlinear behavior when they are under high electromechanical loading. One basic reason for this nonlinearity is the domain switching in the microstructure. In literature some experiments are described which were performed in order to determine non-linear properties of ferroelectric materials. In these measurements, all the experimental set-up are nearly the same. The sample of piezoceramic material is usually subjected to both electric and mechanical loading [2-7]. In experiments especially cyclic electric or mechanical loadings are applied. The magnitude of the loading is slowly increased and decreased at a low frequency in order to observe hysteresis loops. Such hysteresis curves were found both for polarization versus electric field and in form of butterfly loops, in which strain is plotted as a function of the electric field. In these experiments the influence of the composition of some PLZT materials, concentrations of elements, phase changes from tetragonal to rhombohedral or rhombohedral to tetragonal and soft and hard piezoceramic materials are investigated. Micromechanical and phenomenological macroscopic models are two possible approaches to simulate the behavior of piezoceramic materials. Several micromechanical models have been developed for the non-linear behavior of ferroelectric materials. In these models each individual element is randomly oriented and it is assumed that the behavior is that of a microcrystal. The constitutive equations and the switching under the effect of the applied electric field and stress are considered for each element. The contribution of each element to the macroscopic response of a polycrystalline material is calculated by simply averaging over all elements. Hwang et. al. [2] proposed a similar model like that described above. They used a Preisach hysteresis model to describe polarization and strain simulation of each grain of PLZT (Lead lanthanum zirconate titane). The simulations are compared with experiments which were performed with a PLZT 8/65/35. In [8], Michelitsch and Freher present an extended analytical approach of [2]. The model assumes a material that consists of single domain grains which have transversely isotropic microproperties. Macroscopic quantities are derived once more by simply averaging analytically. Chen et al. [9] establish microscopic constitutive relations of single crystal ferroelectric materials by using an internal variable in the constitutive equations and domain volume fractions. The model is developed for predicting the properties of polycrystalline ferroelectrics based on monocrystalline ferroelectrics. Because of differences between monocrystal and poycrystal materials, the simulation of the theory does not match exactly. Hwang and McMeeking [10], presented a simulation for polarization switching to predict electric displacement and strain under electromechanical

3 High Performance Structures and Materials II 79 loading in polycrystalline tetragonal ferroelectric ceramics. In their model, the energy required for switching is approximated by using additional parameters to provide the best fit to experimental data that were measured before. The simulations are carried out for different electrical and mechanical loadings. Still they got reasonable simulations, especially for electric displacement and strain versus electric field curves. However, the simulations are limited in approximating the gradual process of switching. Another micromechanical simulation of tetragonal feerroelectric ceramics is performed by Lu et al. [6] using an orientation distribution function for describing the domain patterns of ferroelectrics in the micromechanical model. This is the basic difference of this model compared to others. Various simulations were performed using combined electric-mechanical loadings. The change of the curves for different stress and electric field values were observed. Another similar approach is used by Chen and Lynch [11]. The model mentions some other phenomena such as differences between tetragonal and rhombohedral structures, different energy levels associated with switching criteria of these structures (90 and 180 for tetragonal, 70.5, and 180 for rhombohedral), interaction effects between grains and phase transitions from rhombohedral to tetragonal, as well as tetragonal to rhombohedral. In addition, they apply a linear saturation model to their calculations in order to match simulations with experimental data. Recently, some macroscopic constitutive models were developed to predict the behavior of ferroelectric materials. These models are usually constructed in a thermodynamic framework. The basic advantage of macroscopic models is that they require less time effort for calculations. Kamlah et al. [12, 13] present a simple phenomenological model for ferroelectric materials to simulate basic dielectric hysteresis, butterfly hysteresis, ferroelectric hysteresis, mechanical depolarization and polarization induced piezoelectricity curves. The model is simple enough to be implemented in a FE-code for the case of uni-axial loadings. They take into account the history dependence of the material behavior by introducing certain internal variables representing the history knowledge. Another different phenomenological model is also presented by Kamlah and Jiang [14, 15]. The model presents macroscopically not only history dependent nonlinearities and dielectric butterfly hysteresis due to switching, but also thermo-electromechanical coupling properties and rate dependent effects in the simulations. McMeeking and Landis [16] proposed a phenomenological constitutive model for ferroelctric switching under multi-axial mechanical and electrical loadings. A constitutive model for non-linear switching of ferroelectric materials is developed by Huber et al. [17]. The model used single crystal constitutive law that is analogous to slip systems of crystal plasticity theory. Although the model captured the smooth shapes of dielectric hysteresis and butterfly hysteresis loops under electromechanical loads and depolarization of polycrystals under compressive loads, the higher computational time is the basic drawback. A more general multi-axial constitutive model based on thermodynamic formulations for ferroelectric ceramics is presented by Landis [18]. The model is derived from Helmholtz free energy equations. In spite of being effective to simulate

4 80 High Performance Structures and Materials II important properties of ferroelectric materials, phenomenological models are too complicated to be solved analytically. Therefore, some numerical methods such as numerical integration are used to model the system. In this paper, a new micromechanical modeling of ferroelectric materials is developed. The basic of this concept is that the nonlinear domain switching occurs with a certain probability. Therefore, the curical point of the model is the propability criteria for domain switching in ferroelectric materials. With such a propability function intergranular effects may be taken into account phenomenologically. The results of the simulations are shown for the dielectric displacement versus electric field hysteresis curves. Barium (Ba) Oxigen (O 2 ) a0 a0 bc a Titanium (Ti) a0 P s = 0 a P s = 0 Figure 1: Lattice structure of cubic and tetragonal elements (BaTiO 3 ). 2 Micromechanical model Piezoceramic materials show a cubic lattice structure for the paralecetric phase which occurs above the Curie temperature and for which the lengths of the edges are the same. By decreasing the temperature below the Curie temperature a transition from the paraelectric phase to the ferroelectric phase occurs for which the PLZT material shows a tetragonal microstructure, see figure 1. In the ferroelectric phase, the net dipole moment in tetragonal lattice element is not zero in contrary to a cubic lattice structure. Therefore, there is a net polarization, the spontaneous polarization, and a net strain, the spontaneous strain which occur both in the microstructure. The spontaneous strain S 0 in the microstructure is coming from the transition from the cubic to the tetragonal structure that depends on the lengths of crystall for both phases (b c and a 0 ) is given by s0 = (bc-a0) a0 (1) A bulk ferroelectric ceramic material is composed of many randomly oriented grains, see figure 2. Each grain has a different spontaneous polarization direction. Therefore, the overall polarization and the strain of a bulk ferroelectric

5 High Performance Structures and Materials II 81 material are approximately zero. Each grain or microstructure s electric displacement and strain values are calculated using constitutive equations, which take into consideration the spontaneous polarization. The macroscopic polarization and the macroscopic strain of the bulk ceramic are calculated by adding the average of the spontaneous values to the results obtained by the linear constitutive equations P= P s + d* σ + ε* E (2) S= S s + s* σ + d* E (3) Figure 2: Randomly oriented grains in a piezoceramic material. In these equations P denotes the average polarization, S the average strain, E the electric field, σ the mechanical stress, P s the spontaneous polarization, S s the spontaneous strain, d the piezoelectric constant, ε the dielectric constant and s is the elastic constant. Like mentioned before, piezoceramic materials show nonlinear properties when they are under high electromechanical loading. The nonlinearity considered here has its origin in the domain switching in the microstructure. Regarding the microstructure of a tetragonal cubic element, there are six possible orientations for the polarization and therefore, two possible types of switching, the 90 and the 180 switching occur. In this paper, a micromechanical model is considered, which assumes a bulk piezoceramic material that has 1000 microcrystalline elements, see figure 3. These elements are assumed to show the characteristics of an individual grain. Each grain has random orientation of the polarization and of the spontaneous strain. Randomness is given by very well known Euler angles between 0 to 2π. A local coordinate systems for each grain and one global coordinate system are introduced in order to transform the values obtained by the calculations in the

6 82 High Performance Structures and Materials II local coordinate system to global coordinates and vice versa. Global coordinates (x g,y g,z g ) T and local coordinates (x l,y l,z l ) T are related by (x g,y g,z g ) T =R(α,β,γ) (x l,y l,z l ) T (4) Figure 3: Micromechanical model in form of 1000 cubic elements. The transformation matrix R(α,β,γ) can be calculated from Euler angles (α, β, γ) by a multiplication of three rotation matrixes Rz(α).Ry(β).Rz(γ) = cosαcosβcosγ-sinαsinγ -cosαcosβsinγ-sinαcosγ -cosαsinβ sinαcosβcosγ+cosαsinγ -sinαcosβsinγ+cosαcosγ sinαsinβ -cosγsinβ sinγsinβ cosβ (5) The authors are well aware that equally distributed Euler-angles do not lead to a uniform distribution of the polarization direction. Nevertheless, equally distributed Euler-angles are widely used for the investigations described in literature. The model assumes that the macroscopic load is equal to the load on each grain of piezoceramic material. In order to match the simulations to the experimental data some parameters such as spontaneous polarization, spontaneous strain, piezoelectric and dielectric constants are chosen from literature. For simplifying the calculations, dielectric and elastic constants are assumed to be isotropic and the same for all grains. On the other hand, the piezoelectric constant is assumed to be transversely isotropic. During the calculation, the value for the dielectric permittivity ε is taken 0.06 µf/m, which is a typical value for the dielectric permittivity for PLZT materials. The value of spontaneous polarization is chosen to be 0.3 C/m 2. At each grain, in addition to the piezoelectric constitutive equations, nonlinear switching models are implemented. The domain switching of each grain is determined by the electromechanical energy criteria. E* P+σ* S 2P 0 *E c (6)

7 High Performance Structures and Materials II 83 In this relation E and σ are electric field and mechanical stress correspondingly, E c is the coercitive electric field, P 0 is the critical value of the spontaneous polarization, Figure 4: Hysteresis curve without probability function. Figure 5: Hysteresis curve with third order probability function. P and S are the polarization and strain change during switching correspondingly. During the simulation, the stress is taken constant and zero. Therefore, on the left side of the inequality, the second term becomes zero. According to the this criteria domain switching occurs if the energy change is higher than a certain critical level. Different energy levels are used for 90 and 180 domain switching during the simulations. Interactions between different grains are assumed to be negligible in these simulations.

8 84 High Performance Structures and Materials II Figure 6: Hysteresis curve with fourth order probability function. Figure 7: Comparison between simulation (fifth order probability function) and measured data. In this paper also a probability density for switching is considered even for energy levels below the critical energy level given in (6). Such an approach is new and was not described in literature before. For a grain the probability to switch the poling direction is taken in form of a polynomial of order n for the ratio of electric field and coercitive electric field p(e) = (E/Ec) n (7) where p is the probability for switching which is varying between 0 to 1 according to the electric field. In simulations, the value for n is chosen as three, four and five. The difference between simulation results with probability criteria

9 High Performance Structures and Materials II 85 and without probability is significant, see figures 4 to 6. A comparison of the simulation with experimental data measured by Lu et al. [6] is given in figure 7. The simulation results obtained with probability criteria look more like curves obtained experimentally than those from simulations without probability function. For the simulations a cyclic, uni-axial electrical field is applied with a maximum absolute value of 2 kv/mm. The starting point for the first cycle is at zero electric field for the unpoled ceramic. Euler transformations and averaging the results of all grains are used to get the macroscopic response of the bulk material to the applied electric loads. 3 Conclusion Nonlinear properties of tetragonal ferroelectric and piezoelectric ceramic materials under high electrical load are simulated in this paper. The simulation is done using a micromechanical approach. The simulations are not only based on a linear constitutive model but also on nonlinear domain switching. Domain switching is not determined by an energy criteria. Instead, switching may occur even for energies which are below energy limits described in literature. This effect is modeled with the help of a probability function. The hysteresis curves which are simulated with this model show better correspondence to the experimental hysteresis curves than simulation results which are only based on an energy criterion. Especially, the agreement is better for electric field levels which are near the coercitive electric field level. This improvement of the simulated curve can be seen both for increasing as well as decreasing electric fields. In future work the focus will be on intergranular effects between neighboring grains. It is known that these effects play an important role in nonlinear effects of piezoceramic materials. References [1] B. Jaffe, W. R. Cook and H. Jaffe, Piezoelectric Ceramics, Academic Press, London and New York, [2] S. C. Hwang, C. S. Lynch & R. M. McMeeking, Ferroelectric/Ferroelastic interactions and a polarization switching model, Acta Metal. Mater., Vol. 43, No. 5, pp , [3] H. Cao and A. G. Evans, Nonlinear deformation of ferroelectric ceramics, Journal of Am. Ceram. Soc., 76, [4], pp , [4] A. B. Schaeufele and K. H. Haerdtl, Ferroelastic properties of lead zirconate titanate ceramics, Journal of Am. Ceram. Soc., 79, [10], pp , [5] C. S. Lynch, The effect of uniaxial stress on the electro-mechanical response of 8/65/35 PLZT, Acta Mater., 44, No. 10, pp , 1996.

10 86 High Performance Structures and Materials II [6] W. Lu, D. -N. Fang, C. Q. Li, K. -C. Hwang, Nonlinear electricmechanical behaviour and micromechanics modelling of ferroelectric domain evolution, Acta Mater., 47, No. 10, pp , [7] D. Zhou, M. Kamlah, D. Munz, Rate dependence of soft PZT ceramics under electric field loading, Smart Structures and Materials., Proceedings of SPIE Vol. 4333, pp , [8] T. Michelitsch, W. S. Kreher, A simple model for the nonlinear material behavior of ferroelectrics, Acta mater., Vol. 46, No. 14, pp , [9] X. Chen, D. N. Fang and K. C. Hwang, Micromechanics simulation of ferroelectric polarization switching, Acta Mater., Vol. 45, No. 8, pp , [10] S. C. Hwang and R. M. McMeeking, The prediction of switching in polycrystalline ferroelectric ceramics, Ferroelectrics, Vol. 207, pp , [11] W. Chen, C. S. Lynch, A micro-electro-mechanical model for polarization switching of ferroelectric materials, Acta Mater., 46, No. 15, pp , [12] M. Kamlah, C. Tsakmakis, Phenomenological modeling of non-linear electromechanical coupling in ferroelectrics, International journal of solids and structures, 36, pp , [13] M. Kamlah, U. Böhle, D. Munz, C. Tsakmakis, Macroscopic description of the non-linear electro-mechanical coupling in ferroelectrics, Smart structures and materials, Proceedings of SPIE Vol. 3039, pp , [14] M. Kamlah, Q. Jiang, A model for PZT ceramics under uni-axial loading, Wissenschaftliche berichte, Forschungszentrum Karlsruhe, Institut für Materialforschung, [15] M. Kamlah, Ferroelectric and ferroelastic piezoceramics-modeling of electromechanical hysteresis phenomena, Continuum mech. thermodyn., 13, pp , [16] R. M. McMeeking, C. M. Landis, A phenomenological multi-axial constitutive law for switching in polycrystalline ferroelectric ceramics, Journal of engineering science., 40, pp , [17] J.E. Huber, N.A. Fleck, C.M. Landis, R.M. McMeeking, A constitutive model for ferroelectric polycrystals, Journal of the mechanics and physics of solids., 47, pp , [18] C.M. Landis, Fully coupled, multi-axial, symetric constitutive laws for polycrystalline ferroelectric ceramics, Journal of the mechanics and physics of solids, 50, pp , 2002.

MODELING AND SIMULATION OF PIEZOCERAMIC MATERIALS USING MICROMECHANICAL APPROACH

MODELING AND SIMULATION OF PIEZOCERAMIC MATERIALS USING MICROMECHANICAL APPROACH European Congress on Computaional Methods in Applied Sciences and Engineering ECCOMAS 2004 P.Neittaanmäki, T.Rossi, K.Majava, and O.Pironneau (eds.) R.Owen and M.Mikkola (assoc. eds.) Jyväskylä, 24-28