Porous piezoelectric materials for energy harvesting

Transcription

1 Porous pieoelectric materials for energ harvesting G. Martine-Auso 1, M. I. Friswell 1, S. Adhikari 1, H. Haddad Khodaparast 1, C. A. Featherston 2 1 Swansea Universit, Ba Campus Fabian Wa, Crmln Burrows, Swansea SA1 8EN, Wales, United Kingdom @swansea.ac.uk 2 Cardiff Universit, School of Engineering, Queen s Buildings, The Parade, Cardiff CF24 3AA, Wales, United Kingdom Abstract In this paper, we assess the energ harvesting capabilities of porous pieoelectric material under harmonic ecitation and investigate the advantages of functionall grading the air inclusions. A cantilever beam energ harvester with base ecitation is used to demonstrate the effects of porosit on the power generated. A homogeniation step using the analtical Mori-Tanaka approach is performed initiall to reduce the computational requirements. This homogeniation will estimate the material properties for different levels of porosit. An Euler-Bernoulli beam model is used to efficientl estimate the power generated for a pieoelectric sensor with uniform properties. A 2D finite element model is then developed to verif the beam model; this detailed model ma be used to anale harvesters where the porosit varies through the thickness or along the length of the beams. An optimiation is performed, focusing on the impact of the percentage of inclusions on the energ harvesting efficienc. 1 Introduction Pieoelectric materials generate an electrical field when a strain is applied to them, which is called the direct pieoelectric effect. These materials also ehibit the opposite effect, where a strain arises when an electrical field is applied, and this is called inverse pieoelectric effect. These materials are widel used in man applications, for eample inkjet printers, sonars, heart monitors, tennis racquets, hdrophones or air bag sensors, either as actuators (direct effect) or as sensors (inverse effect). A more recent application is energ harvesters using a vibration source, that are used to power small devices or recharge batteries. A. Ertuk and D.J. Inman [1 3] studied pieoelectric cantilever bimorph beams for energ harvesting, both theoreticall and eperimentall. A. Sodano [4 6] gave an etensive reviews of the applications of pieoelectric materials to energ harvesting. M.I. Friswell and S. Adhikari eplored the possibilities of pieoelectric devices to harvest energ from non-linear vibration [7] and from broadband ecitation [8]. For energ harvesting applications, two coefficients are used to measure the capabilit of a pieoelectric energ harvester; the electromechanical coupling and the capacitance [2]. These two coefficients depend on the material properties and the device geometr. The electromechanical coupling coefficient measures the capabilit of the harvester to convert strain to electrical field, and hence is based on the stiffness and pieoelectrical properties of the materials. Most approaches have focused on maimiing the strain in the pieoelectrical material through a careful optimiation of the geometr [9, 1]. Some researchers have studied the impact of the electrical circuit configuration on the power output [11, 12], but few researchers have focused on reducing the capacitance. The capacitance depends on the dielectric characteristics of the material and the geometr of the device. The dielectric coefficients ma be reduced b miing the 75

2 76 PROCEEDINGS OF ISMA216 INCLUDING USD216 pieoelectric material with other materials which lower permittivit to create new composite materials. A significant reduction for porous pieoelectric material with inclusions of air, which has ver low dielectric properties was reported in [13]. Porous pieoelectric materials have been studied b a number of authors. J.F. Li et al. [14] studied this material eperimentall for actuator applications. E. Roncari et al. [15] and J. I. Roscow et al. [16] reviewed the manufacturing process of porous pieoelectric materials. M.L. Dunn and M. Taa[17] applied homogeniation mean field theor to analticall predict the mechanical, pieoelectric and dielectric properties of porous pieoelectric materials. There is little understanding of the effects of porous pieoelectric materials in energ harvesting. This paper aims to fill this gap in knowledge through a stud of porous pieoelectric bimorph beams under harmonic ecitation. Using the mean-field homogeniation approach, the effective material properties are calculated for different percentage of porosit. These properties are then used to model a pieoelectric cantilever beam harvester under harmonic base ecitation, which drives a load resistor. The voltage and power are calculated for various levels of porosit, different ecitation frequencies and different load resitance values. The structure of this paper is summaried as follows. In section 2 the modeling approach is described; this section is divided into three parts, according to the different methods used. Section 2.1 introduces the Mori- Tanaka method to homogenie composite materials such as the porous pieoelectric material. Section 2.2 gives the analtical solution for a bimorph beam based on Euler-Bernoulli beam theor. Finall, a finite element model (FEM) is used to verif the beam model in section 2.3; this model will enable functionall graded pieoelectric material to be analed and optimied. The results for an eample harvester device are given in section 3 and the main conclusions are summaried in section 4. 2 Modelling of a cantilever beam energ harvester The constitutive equations for linear pieoelectricit can be derived b considering an electric enthalp function H defined, for the linear static case without bod charge or forces, as [18, 19] ( 1 H(ϵ, E) = 2 ϵ ijcijmnϵ E mn 1 ) 2 E ikine ϵ n + E n e nij ϵ ij dv (1) V In this equation, the independent variables are the elastic strain ϵ mn and the electric field E n. On the right side of the equation, Cijmn E are the elastic constitutive constants measured at constant electric field, e nij are the pieoelectric constants (measured at a constant strain or electric field) and kin ϵ are the dielectric constants measured under constant strain. Differentiating this equation respect to the independent variables gives the constitutive equations of pieoelectricit as σ ij = and D i = H(ϵ, E) = C ijmn ϵ mn + e nij E n ϵ ij (2) H(ϵ, E) = e imn ϵ mn k in E n E i (3) where the stress is σ ij and the electric displacement is D i. The electrical field is related to the voltage through a gradient, equation (6), and the electrical field is related to the electrical displacement through equation (5): ϵ mn = 1 2 (u m,n + u n,m ) (4) D m = k mn E n (5) and E m = ϕ,m (6) From equation (2), in the linear case, the stress applied to a pieoelectric material is converted to elastic deformation and an electrical field proportional to the pieoelectric constitutive matri. This electrical field will

3 DYNAMICS OF ENERGY HARVESTERS 77 also create a gradient through equation (6), which will generate the voltage from the energ harvester. Interestingl, this electrical field also provokes energ loss, since equations (3) and (5) shows that the electrical field will create a self-induced electrical field with opposite sign, proportional to the permittivit constants. Hence the voltage output from the energ harvester is maimied b reducing the permittivit constants and increasing the pieoelectric constants. 2.1 The Mori-Tanaka method The porous pieoelectric material is composed b two phases; air and pieoelectric material. The pieoelectric material is normall lead irconate titanate (PZT) or Barium Titanate. For each phase the material constants are well known, but the set of homogenied material constants must be calculated for the composite. One approach is to solve the detailed finite element model for specific percentages of random inclusions. However, this approach is computationall epensive, and requires man elements to obtain an accurate model. Moreover, for each percentage of material, the model must be re-calculated. An alternative approach is to homogenie the material using analtical methods. One of the most well-known and validated approaches is the Mori-Tanaka method which is based on mean-field homogeniation theor. This method improves the Eshelb solution [2] given for ellipsoidal inclusions in elastic mediums, and was epanded b Y. Benveniste [21] to include composite materials and later b M. L. Dunn and M. Taa [22] to electromechanical fields. The approach ma be used in multi-field phsics analsis, for eample for general composite materials [23 26] and for porous pieoelectric materials [17]. To perform a Mori-Tanaka homogeniation, the calculation of the Eshelb tensor is required. The procedure to obtain this tensor is comprehensivel detailed in [27] and hence is not eplained here. In the Mori-Tanaka method, each inclusion with properties E I, behaves as an isolated inclusion, embedded in an infinite matri with properties E M, that is loaded remotel b an applied strain. Hence each inclusion is subjected to the averaged stress fields acting on it from all of the other inclusions, through the superposition of stresses. The homogeniation procedure of this method is briefl summaried. First an influence tensor has to be calculated for ever phase r (A I,r ) and percentage. This concentration tensor is assumed to be equal to the relation between the strain in the inclusion and the strain in the matri [23]. Thus This concentration tensor is written in terms of the Eshelb tensor, S, as A I,r = [I + S ( E M) 1 ( E I,r E M)] 1 E I,r = A I,r E M (7) These concentration tensors are then averaged to obtain the general influence tensor, A I,r (MT) = c I,r I + c M (A I,r ) 1 + N j=1 c I,r A I,j (AI,r Finall, the effective electro-elastic material tensor (E ) is obtained using and E = N c r E r A r = E M + r=1 E (MT) = EM + ) 1 1 ( ) A I (MT) (8) (9) N c r ( E I E M) A r (1) r=2 N c I,r ( E I,r E M) A I,r (MT) (11) r=1 This method is self-consistent, since the inverse of the electromechanical matri E is equal to the compliance electromechanical matri F.

4 78 PROCEEDINGS OF ISMA216 INCLUDING USD Analtical Modelling of the Energ Harvester The response of the energ harvester is predicted first b an analtical solution. A cantilever beam energ harvester is proposed; man authors have chosen this tpolog [1, 7, 8, 11, 12] due to the high strains developed at the clamped end, and because it is eas to build and test. Furthermore, analtical solutions for beam sstems are highl developed, allowing representative models to be implemeted relativel easil. The proposed cantilever beam is composed of two pieoelectric laers with an elastic support laer in between, and clamped at the right side, as shown in fig. 1. PZT Polariation Y v(t) R X PZT Polariation Figure 1: Schematic view of the cantilever beam energ harvester with the circuit configuration. The beam is smmetric about its neautral ais, and its thickness is small compared to its length. An Euler- Bernoulli beam model is considered with a resistor connected in series to the electrodes on the top and bottom pieoelectric material surfaces. Following the approach presented in [2], the eternal circuit admittance, which is inversel proportional to the resistance, is related to the pieoelectric beam through the integral form of Gauss s law: d D n da = v(t) (12) dt R where R is the load resistance. Substituting equation (3) in equation (12), one obtains A k 33 bl dv(t) + v(t) 2h p dt R + e 31 (h p + h e ) L b 2 3 w(, t) 2 d = (13) t In this equation, k 33 is the permittivit coefficient and e 31 is the pieoelectric coefficient. The geometrical dimensions b,l,h p,h e are the beam width, beam length, the pieoelectric laer thickness, and the elastic laer thickness. The displacement along the beam is represented b w; this is relative to the base motion of the cantilever beam. The voltage across the load resistor is v(t). In equation (13) the unknown variables are v(t) and the displacement w. Using a linear modal decomposition analsis, the displacements ma be epanded as w(, t) = ϕ n () η n (t) (14) n=1 where ϕ n () is the nth mass-normalied mode shape of the cantilever beam. The mechanical modal coordinate for the nth mode is η n. Assuming a harmonic base ecitation, the voltage is harmonic and given b v(t) = V (ω) e jωt. Then, from equations (13) and (14), the voltage V (ω) is [2] V (ω) = ω 2 W n=1 1 R + jωceq + jωθ n σ n ω 2 n ω 2 + j2ζ n ω n ω n=1 jωθ 2 n ω 2 n ω 2 + j2ζ n ω n ω (15)

5 DYNAMICS OF ENERGY HARVESTERS 79 where n is the mode number, ω is the frequenc of the base ecitation, w n is the natural frequenc of the n th mode, W is the amplitude of the base ecitation, ζ n is the modal damping for mode n, R is the load resistance of the device connected to the energ harvester, and j is the unit imaginar ( 1). The parameters σ n, θ n, C ep are coefficients related to the inertia of the beam, the electro-mechanical coupling coefficient and the electrical capacitance respectivel. For a cantilever beam without tip mass, and with the pieoelectric laers covering the whole length of the beam and connected in series, these coefficients are σ n = m where m is the mass per unit length of the beam. L ϕ n ()d (16) ( ) hp + h e dϕn θ n = e 31 b 2 d (17) =L C eq = ks 33 bl 2h p (18) The instantaneous power generated b the harvestor is obtained as P = V I = V 2 /R, where I is the current through the resistor. An analtical stud is performed for harvesters with pieoelectric laers that have uniform properties, but for different percentages of inclusions. This will give some insight into the effect of porosit of the pieoelectric material on the energ harvester performance. Although the objective, namel to maimise the harvested power, is ver clear, choosing the constraints to appl to the harvester sstem is not so straightforward. Hence three models will be considered, which show the effect of the percentage of pieoelectric material for a specific constraint (geometrical properties, mass and frequenc). Unless otherwise specified, the geometrical properties in table 1 are used, and the material properties are given in equation (19). The three models are: Model A. Thickness constant. The geometr is constant, and onl the percentage of inclusions in the pieoelectric material is varied. Model B. Mass constant. The mass of pieoelectric material remains constant. As the percentage of material in the pieoelectric laers decreases, the thickness increases at the same rate to keep the pieoelectric material mass constant. Model C. The first mechanical natural frequenc is kept constant. The mass and stiffness of the beam is changed b the percentage of the pieoelectric material, and hence the first natural frequenc is also changed. The thickness is modified b an iterative solver to fi the first natural frequenc (to 185H in the eample). 2.3 Finite Element Modelling In this section the finite element model (FEM) used is eplained briefl. The FEM ma be used to validate the beam model, but will also allow the distribution of pieoelectric material to be optimied b varing the porosit along the length and through the thickness of the beam. The model uses Matlab to launch the FEM model code in ANSYS. The model uses 2D elements to reduce the computational cost, and the element tpe used for the pieoelectric material is PLANE223. This element has 8 nodes and a quadratic displacement behavior. For the non-pieoelectric elements, the element tpe is PLANE183, which also has 8 nodes and a quadratic displacement behavior. The FE model is coded using the APDL programming language, a proprietar language used in ANSYS. The user chooses the percentage at a reduced number of sections of the beam; the Matlab script then interpolates these values to give a fine distribution, using a cubic spline with ero slope at the beginning

6 71 PROCEEDINGS OF ISMA216 INCLUDING USD216 and end of the beam. This distribution is discretied into elements with constant properties given at their central point. The nodes, element and material properties are then written to input files for ANSYS. After ANSYS has solved the model, the APDL code prints the results to different output files which are read b the Matlab script. 3 Results The geometrical properties of the cantilever beam and the material properties of the elastic support, are detailed in table 1. The material used in the simulations is PZT-5A, whose properties are given b [2] Geometr Elastic Material Properties Beam Length (mm) 3 Elastic Modulus (GPa) 7 Pieoelectric Thickness (mm).15 Poisson s ratio.3 Elastic Laer Thickness (mm).5 Analsis Parameters Modal Damping Ratios.1,.12,.3,.59,.97 Load Resistance (Ω) 1, 1, 1, 1, 1, 1, 1, 1, 1 Table 1: Geometrical properties of the beam, material properties of the elastic support material and analsis parameters. E M = Units = Smmetric Smm C (GPa) e T (C/m 2 ) e (C/m 2 ) k T /k Densit = 775 kg/m 3 k = pf/m (19) At an initial step, the homogenied material properties are obtained using the Mori-Tanaka method. The resulting homogenied properties of the porous material for 5% to 1% of pieoelectrical material are shown in fig. 2. This method gives smooth and consistent results, close to the results given in [17]. The dnamic behavior of different beam models with different levels of of porosit is analed for harmonic base ecitation with an amplitude of 1 mm and in the frequenc range of 1H to 1kH. The frequenc response for model A, which has constant percentage of inclusions along the length, is shown in fig. 3. Under 4% of matri (pieoelectric) material, the porous structure is not stable due to cracks and the lack of a consistent structure [13]; hence, the percentages analed are between 5% and 1%. The effect of the porosit is to reduce the resonance frequenc, and also to reduce the maimum displacement. The reduced displacement will lead to a reduced output voltage. This is shown in fig. 4, which gives the voltage FRF for a range of frequencies close to the first mode for different constant percentages of PZT. The shift in the resonance frequencies is about 3H and the reduction in voltage between the non-porous case and the 5% porous case is about 42.7%.

7 DYNAMICS OF ENERGY HARVESTERS 711 Mechanical Coefficients (1 11 ) Pieoelectrical Coefficients (1 1 ) ( ) Permittivit Coefficients k k 4 2 S 11 S 22 S 12 S 13 S 44 S d 12 d 32 d 51 1,8 1,6 1,4 1,2 k 11 k , % 8 % 1 % Percentage of PZT material 6 % 8 % 1 % Percentage of PZT material 6 % 8 % 1 % Percentage of PZT material Figure 2: The estimated material coefficients obtained using the Mori-Tanaka method. Voltage FRF Voltage FRF Percentage 1% Percentage 9% Percentage 8% Percentage 7% Percentage 6% Percentage 5% , 2, 3, 4, 5, 6, 7, 8, 9, 1, Frequenc (H) Figure 3: Frequenc response for different percentages of pieoelectric material. The results for the different simulated models are shown in figs. 5, 6 and 7. Three results are shown as surfaces. The first is the maimum voltage near the first mode as a function of load resistance and pieoelectric material percentage. The second is the maimum power harvested around the first mode. The third gives the frequenc where this maimum power is achieved. The results for Model A are shown in figures 5(a), 5(b) and 5(c). The voltage generated b the energ harvester decreases smoothl with the percentage of pieoelectric material. The reduction in the quantit of the PZT material leads to a reduction in all of the main parameters, namel frequenc, voltage and power. The results for Model B are shown in figure 6(a), 6(b) and 6(c). This model keeps the mass of the pieoelectric material constant as the percentage of pieoelectric material varies. Hence, to keep the mass constant, the thickness of the pieoelectric laers have to be increased in the same ratio to the porosit increase. The results show a significant increase in the voltage generated, as well as the power harvested. Interestingl, the resonance frequenc in these cases tends to increase as the the porosit increases. This trend is in contrast to that shown b the model A, and is due to the increasing thickness of the harvester, which moves pieoelectric material further from the neutral ais, hence increases the strain in the pieoelectric laer. Higher strains in the pieoelectric laers, contribute to an increase in the electrical field generated b the pieoelectric effect,

8 712 PROCEEDINGS OF ISMA216 INCLUDING USD216 Voltage FRF Voltage FRF Percentage 1% Percentage 9% Percentage 8% Percentage 7% Percentage 6% Percentage 5% Frequenc (H) Figure 4: The voltage frequenc response around the first mode for different percentages of pieoelectric material. and therefore the voltage. The increasing frequenc means that the stiffness variation is lower than the mass variation, due to the non-linear variation in the electromechanical properties calculated b the Mori-Tanaka method. In contrast, the mass changes linearl with the percentage of pieoelectric material, and has a bigger impact on the resonance frequenc. The results for Model C are shown in figures 7(a), 7(b) and 7(c). The first mechanical natural frequenc of the energ harvester is fied at 185H. To obtain this frequenc, the thickness of the pieoelectric laer is varied. The trends for this model are quite similar to the ones obtained for Model A, with decreasing frequenc, voltage and frequenc for increasing porous inclusion percentage , % 8 % % % 8 % 1 % % 8 % 1 % (a) Voltage (b) Power (c) Frequenc Figure 5: Results for Model A: Thickness constant. The -ais is the percentage pieoelectric material and -ais is the load resistance: (a) the maimum voltage of the energ harvester under harmonic ecitation for frequencies around the first resonance; (b) The maimum power around the first resonance; (c) the frequenc for maimum power. 4 Conclusions The dnamic behaviour of a porous pieoelectric energ harvester was analsed. The dnamic response has been predicted using an analtical model of a simple bimorph cantilever beam. The modeling process

9 DYNAMICS OF ENERGY HARVESTERS % 8 % % % 8 % 1 % % 8 % 1 % (a) Voltage (b) Power (c) Frequenc Figure 6: Results for Model B: Mass constant. The -ais is the percentage pieoelectric material and - ais is the load resistance: (a) the maimum voltage of the energ harvester under harmonic ecitation for frequencies around the first resonance; (b) The maimum power around the first resonance; (c) the frequenc for maimum power. 4, , % 8 % % % 8 % 1 % % 8 % 1 % (a) Voltage (b) Power (c) Frequenc Figure 7: Results for Model C: Frequenc constant. The -ais is the percentage pieoelectric material and -ais is the load resistance: (a) the maimum voltage of the energ harvester under harmonic ecitation for frequencies around the first resonance; (b) The maimum power around the first resonance; (c) the frequenc for maimum power. requires a homogeniation approach, which in this case have been performed using the Mori-Tanaka method. The results shows a general decrease of the material properties and energ harvested b the device with a decrease in the mass of the pieoelectric material. When the mass of the pieoelectric material is fied for each level of porosit (Model B) the results shows a significant improvement in the energ harvested. These results show that using porous material to optimie the distribution of pieoelectric material can harvest more energ than a non-porous material. The present paper is part of ongoing research at Swansea Universit investigating the applications of porous materials for energ harvesting. In future research the impact of the distribution of the porosit along the thickness and length will be studied using the FE model presented. Acknowledgements The authors acknowledge the financial support from the Sêr Cmru National Research Network and Swansea Universit through a Postgraduate Scholarship.

10 714 PROCEEDINGS OF ISMA216 INCLUDING USD216 References [1] A. Erturk, D. J. Inman. An eperimentall validated bimorph cantilever model for pieoelectric energ harvesting from base ecitations. Smart Materials and Structures, vol. 18, no. 2 (29), p [2] A. Erturk, D. J. Inman. Pieoelectric energ harvesting. John Wile & Sons (211). ISBN [3] A. Erturk, D. J. Inman. On mechanical modeling of cantilevered pieoelectric vibration energ harvesters. Journal of Intelligent Material Sstems and Structures (28). [4] H. A. Sodano, D. J. Inman, G. Park. A review of power harvesting from vibration using pieoelectric materials. Shock and Vibration Digest, vol. 36, no. 3 (24), pp [5] H. A. Sodano, D. J. Inman, G. Park. Comparison of pieoelectric energ harvesting devices for recharging batteries. Journal of Intelligent Material Sstems and Structures, vol. 16, no. 1 (25), pp [6] S. R. Anton, H. A. Sodano. A review of power harvesting using pieoelectric materials (23 26). Smart Materials and Structures, vol. 16, no. 3 (27), p. R1. [7] M. I. Friswell, S. F. Ali, O. Bilgen, S. Adhikari, A. W. Lees, G. Litak. Non-linear pieoelectric vibration energ harvesting from a vertical cantilever beam with tip mass. Journal of Intelligent Material Sstems and Structures, vol. 23, no. 13 (212), pp [8] S. Adhikari, M. I. Friswell, D. J. Inman. Pieoelectric energ harvesting from broadband random vibrations. Smart Materials and Structures, vol. 18, no. 11 (29), p [9] C. J. Rupp, A. Evgrafov, K. Maute, M. L. Dunn. Design of pieoelectric energ harvesting sstems: a topolog optimiation approach based on multilaer plates and shells. Journal of Intelligent Material Sstems and Structures, vol. 2, no. 16 (29), pp [1] V. R. Challa, M. Prasad, Y. Shi, F. T. Fisher. A vibration energ harvesting device with bidirectional resonance frequenc tunabilit. Smart Materials and Structures, vol. 17, no. 1 (28), p [11] E. Lefeuvre, D. Audigier, C. Richard, D. Guomar. Buck-boost converter for sensorless power optimiation of pieoelectric energ harvester. Power Electronics, IEEE Transactions on, vol. 22, no. 5 (27), pp [12] G. K. Ottman, H. F. Hofmann, A. C. Bhatt, G. A. Lesieutre. Adaptive pieoelectric energ harvesting circuit for wireless remote power suppl. Power Electronics, IEEE Transactions on, vol. 17, no. 5 (22), pp [13] C. R. Bowen, A. Perr, A. Lewis, H. Kara. Processing and properties of porous pieoelectric materials with high hdrostatic figures of merit. Journal of the European Ceramic Societ, vol. 24, no. 2 (24), pp [14] J. F. Li, K. Takagi, M. Ono, W. Pan, R. Watanabe, A. Almajid, M. Taa. Fabrication and evaluation of porous pieoelectric ceramics and porosit graded pieoelectric actuators. Journal of the American Ceramic Societ, vol. 86, no. 7 (23), pp [15] E. Roncari, C. Galassi, F. Craciun, C. Capiani, A. Piancastelli. A microstructural stud of porous pieoelectric ceramics obtained b different methods. Journal of the European Ceramic Societ, vol. 21, no. 3 (21), pp [16] J. I. Roscow, J. Talor, C. R. Bowen. Manufacture and characteriation of porous ferroelectrics for pieoelectric energ harvesting applications. Ferroelectrics, vol. 498, no. 1 (216), pp

11 DYNAMICS OF ENERGY HARVESTERS 715 [17] M. L. Dunn, M. Taa. Electromechanical properties of porous pieoelectric ceramics. Journal of the American Ceramic Societ, vol. 76, no. 7 (1993), pp [18] Q. Qin. Advanced mechanics of pieoelectricit. Springer Science & Business Media (212). [19] E. P. Eer Nisse. Variational method for eletcroelasticvibration analsis. The Journal of the Acoustical Societ of America, vol. 4, no. 5 (1966), pp [2] J. D. Eshelb. The determination of the elastic field of an ellipsoidal inclusion, and related problems. In Proceedings of the Roal Societ of London A: Mathematical, Phsical and Engineering Sciences, vol The Roal Societ (1957), pp [21] Y. Benveniste. A new approach to the application of mori-tanaka s theor in composite materials. Mechanics of materials, vol. 6, no. 2 (1987), pp [22] M. L. Dunn, M. Taa. An analsis of pieoelectric composite materials containing ellipsoidal inhomogeneities. In Proceedings of the Roal Societ of London A: Mathematical, Phsical and Engineering Sciences, vol The Roal Societ (1993), pp [23] B. Klusemann, B. Svendsen. Homogeniation methods for multi-phase elastic composites: Comparisons and benchmarks. Technische Mechanik, vol. 3, no. 4 (21), pp [24] S. Kurukuri, S. Eckardt. A review of homogeniation techniques for heterogeneous materials. Advanced Mechanics of Materials and Structures, Graduate School in Structural Engineering, German (24). [25] O. Pierard, C. Friebel, I. Doghri. Mean-field homogeniation of multi-phase thermo-elastic composites: a general framework and its validation. Composites Science and Technolog, vol. 64, no. 1 (24), pp [26] T. Te Wu. The effect of inclusion shape on the elastic moduli of a two-phase material. International Journal of Solids and Structures, vol. 2, no. 1 (1966), pp [27] Y. Mikata. Determination of pieoelectric eshelb tensor in transversel isotropic pieoelectric solids. International Journal of Engineering Science, vol. 38, no. 6 (2), pp

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