Studies on effective fiber and matrix poling characteristics of 1-3 piezoelectric composites

Transcription

1 INTERNATIONAL JOURNAL OF STRUCTURAL CHANGES IN SOLIDS Mechanics and Applications Volume 3, Number 3, 211, pp Studies on eective iber and matrix poling characteristics o 1-3 piezoelectric composites M. Sakthivel, A. Arockiarajan a Department o Applied Mechanics, Indian Institute o Technology Madras, Chennai 636, India. Abstract An analytical model is developed to evaluate the perormance o 1-3 piezoelectric composite where both matrix and iber materials are piezoelectrically active. A parametric study is conducted to investigate the eects o variations in the poling characteristics o the iber and matrix phase on the overall thermo-electro-mechanical behavior o a 1-3 piezocomposite. The igures o merit FOM o the 1-3 composite as a transducer or underwater acoustics and biomedical imaging applications has been analyzed. The proposed model is capable o predicting the eective properties o the composite subjected to thermo-electro-mechanical loading conditions or dierent poling directions. The predicted variations in the eective elastic, piezoelectric and dielectric material constants are nonlinear in nature or various iber volume raction. The inluence o thermal eects on eective properties o the composite has been identiied. The analytical results show that an appropriate selection o the poling characteristics o the individual iber and matrix phases could lead to the development o a piezocomposite with signiicant eective properties Key words: Pyroelectricity; Thermal eects; Eective properties; Electromechanical coupling actor. 1. Introduction Piezoelectric materials represent a popular class o active materials which transer energy between the mechanical domain and electrical domain. Monolithic or bulk Piezoelectric materials have been widely used as actuators and sensors, given their high electromechanical coupled perormances. However, bulk piezoelectric materials have certain limitations such as brittle behavior, high acoustic impedance and are very sensitive to cracking; Nelson 22. Piezoelectric composite materials have been introduced where the bulk piezoelectric material in the orm o ibers, is combined with a ductile polymer matrix; reer Newnham et al Piezoelectric composites have been abricated in several orms based on iber and matrix connectivity; or instance -3, 1-3 and piezoelectric ceramic/polymer composites exhibit better characteristics over bulk piezoelectric materials, such as high piezoelectric constant, minor lateral electromechanical coupling, low acoustic impedance, broad bandwidth, and so on; Uchino 2, Topolov and Bowen 29. Hence, by selecting dierent piezoelectric ceramics and polymers or dierent proportions and spatial scales o ceramics, the piezocomposite properties yield better technological solutions then can meet the requirements o some special devices such as medical diagnostic ultrasonic transducers, hydrophones Klicker et al. 1981, Avellaneda and Swart In the literature, several studies have been presented on developing analytical and numerical models or predicting the eective electromechanical response o composites. A simple analytical model is developed by Newnham et al based on series and parallel connectivity or piezoelectric composites. This model concludes that the composite models can be used or interpreting the properties o a single phase materials as aarajan@iitm.ac.in

2 2 International Journal o Structural Changes in Solids well as composites. The dilute, sel-consistent, Mori-Tanaka MT and dierential micromechanics theories are extended to predict the eective electromechanical properties o piezoelectric composites and compared with existing experimental data Dunn and Taya An extension o MoriTanaka and Sel-consistent approaches are proposed by Odegard 24. This work oers a comprehensive comparison o the our modeling approaches or a wide range o matrix and reinorcement electromechanical properties, reinorcement geometry and reinorcement volume raction. Kar-Gupta and Venkatesh 27 presented an analytical model using the equivalent layer composite model where both the matrix and iber phases are piezoelectrically active. Recently, the inluence o porosity in the polymer matrix as well as passive and active matrix on the perormance o 1-3 piezoelectric composites was addressed based on a micromechanics based approach Della and Shu 28. Similarly, several work based on inite element FE ormulation have been presented on the literature to understand the constitutive behavior o piezoelectric composites wherein the interace eects are addressed; reer Poizat and Sester 1999; Pettermann and Suresh 25; Berger et al. 27. The piezocomposites are used in many applications where the impact o thermal eects are in a considerable range and the determination o eective properties is o signiicant importance under thermal eects Tauchert Since the usage o piezocomposite under thermal environment is unavoidable in the current status, it is important to study on the eective thermo-electro-mechanical properties on 1-3 piezocomposite. Though determination o eective electro-mechanical properties o 1-3 composite have been reported in the literature, authors are not aware o any published work in determination o eective thermo-electro-mechanical properties o such 1-3 composites except Kumar and Chakraborty 29. Hence, this work ocuses on thermo-electromechanical coupling studies on composites. Secondly, the prediction o eective properties on piezoelectric composites are reported in the literature, but there are very ew contributions to show what will happen to the eective piezoelectric behavior o the composites i the piezoelectric ceramic PZT rods are not perectly aligned with the loading direction; reer Fig. 2- Class V material. Nan et al. 2 work ocuses on the eective properties o 1-3 type piezoelectric composites with various orientations o lead zirconate titanate rods/ibers aligned in an epoxy matrix using the Green s unction technique. Due to manuacturing diiculties, the iber rod alignment may not be parallel to the loading direction in the 1-3 piezocomposites. This issue motivates us to study the eective properties o composites under dierent iber poling direction. The piezocomposite with two piezoelectric phases iber and matrix are active phases can be used to abricate novel structure transducers such as two intertwined arrays using the ceramic phase to transmit and using the piezo-polymer matrix to receive. Also, the piezoelectric coeicients o the ceramic and polymer are o opposite signs and signiicantly dierent Taunaumang et al But, it may be possible to pole the two phases in opposite directions, that will give enhanced coupling coeicient due to the add-on contribution rom active iber as well as matrix. This problem deinition motivates to do the parametric study on the eective properties o composites under dierent matrix poling directions with respect to iber directions. By having this three problem in-hand, the overall objectives o the present study are: 1. to develop an analytical model to capture the eective thermo-electro-mechanical response o 1-3 piezoelectric composites where both the iber and matrix are transversely isotropic and piezoelectrically active. 2. to predict and compare the undamental elastic, dielectric and piezoelectric material constants o 1-3 piezoelectric composites or dierent iber and matrix poling direction. 3. to study the inluence o ceramic base volume raction on the variation o eective thermal and pyroelectric properties. 4. to perorm a parameter study to examine the inluence o ceramic base volume raction on the perormance o 1-3 type composite or underwater and biomedical imaging transducer applications and compare the perormance or, 9, 18 and 27 matrix poling direction Class I - IV and or, 3, 45, 6 and 9 iber poling direction Class V; see Fig. 2. The outline o the paper is as ollows: the model ormulation is discussed in Section 2 based on the constitutive behavior o piezoelectric composite, assumptions and validity o the model, eective parameter expressions o the 1-3 composite and the perormance ormulation o 1-3 piezocomposite or iber and matrix poling directions. Aspects o the related implementation, analytical treatment and numerical examples are discussed in Sections 3. Finally, a short summary concludes the paper, see Section 4.

3 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites 3 Piezoceramic rod iber Polymer matrix Figure 1: Schematic o a 1-3 piezoelectric composite Class I Class II Class III Class IV Class V - Fiber poling direction - Matrix poling direction Figure 2: Illustration on classiication o 1-3 piezocomposites based on the relative orientation o the poling directions o the iber and matrix phases Class I-V 2. Model ormulation The piezoelectric composite materials can be classiied into ten dierent connectivities, based on the iber and matrix phases in zero, one, two, or three dimensions Newnham et al The 1-3 piezoelectric composites consist o the piezoelectric rods continuous connectivity in one dimension which are embedded into the active matrix that has continuous connectivity along all three dimensions, reer in ig. 1. Two distinct choices or the poling characteristics o the iber and matrix phases poled in a direction parallel to iber axis, and poled in a direction transverse to the iber axis result in our unique piezocomposite classes Fig. 2, described as ollows: I Fiber and matrix phases are poled in longitudinal directions both in +3 axis that are aligned and parallel with the iber axis. II Fiber is poled in a longitudinal direction +3 axis and matrix phase is poled in a transverse direction +1 axis. III Fiber and matrix phases are poled in longitudinal directions +3 axis and -3 axis that are aligned and antiparallel with the iber axis. IV Fiber is poled in a longitudinal direction +3 axis and matrix phase is poled in a transverse direction -1 axis. V Fiber is oriented with some angle θ about the longitudinal direction +3 axis and matrix phase is poled in a longitudinal direction +3 axis.

4 4 International Journal o Structural Changes in Solids Constitutive relation o piezoelectric composite Since the 1-3 composite can be treated as bulk piezoelectric ceramics, they have identical linear constitutive equations as piezoelectrics, which can be written as σ ij = C E ijkl ε kl e ijk E k β ij θ D i = e ikl ε kl + κ ε ij E j + P i θ 1 where σ ij, ε ij, E i and D i are the stress tensor, stain tensor, electric ield vector and electric displacement vector, respectively. C ijkl, e ijk, κ ij, β ij = C ijkl α kl and P i are the elastic stiness tensor, piezoelectric tensor, permittivity tensor, thermal coeicient tensor, and pyroelectric vector, respectively. θ is the change in temperature. The superscripts E and ε indicate that the elasticity and permittivity constants are determined under the conditions o zero or constant electric ield and strain, respectively. Based on the representation o Barnett and Lothe 1997, Della and Shu 28, Eq. 1 can be represented in a single constitutive relation as Σ ij = E ijmr Z MR 2 where Σ ij = E ijmr = { σij J = 1, 2, 3 D i J = 4 C ijmn J, M, R = 1, 2, 3 e nij J, R = 1, 2, 3; M = 4 e imn J = 4; M, R = 1, 2, 3 κ in J, M = 4; R = 1, 2, 3 β ij M, R = 4; J = 1, 2, 3 P i J, M, R = 4 ϵ mn M, R = 1, 2, 3 Z MR = E n M = 4; R = 1, 2, 3 θ M, R = here, the lowercase subscripts are summed over the range o 1-3, whereas the upper case subscripts are the assumed values assigned against the particular variable. 1-3 piezoelectric composite resembles similar to the transversely isotropic piezoelectric constitutive equations, Eq. 2 can be written in matrix orm Voigt s notation as σ x 11 σ x 22 σ x 33 σ x 23 σ x 13 σ x 12 D x 1 D x 2 D x 3 = C x 11 C x 12 C x 13 e x 31 β x 11 C x 12 C x 11 C x 13 e x 31 β x 22 C x 13 C x 13 C x 33 e x 33 β x 33 C x 44 e x 15 C x 44 e x 15 C x 66 e x 15 κ x 11 P x 1 e x 15 κ x 11 P x 2 e x 31 e x 31 e x 33 κ x 33 P x 3 ε x 11 ε x 22 ε x 33 2ε x 23 2ε x 13 2ε x 12 E x 1 E x 2 E x 3 θ x 6 where the superscript x reers to denote the composite as c, the iber as, and the matrix as m.

5 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites 5 b a V V m c V V m Figure 3: Representation o a 1-3 piezoelectric composite as equivalent layered composite with proportional iber and matrix volume ractions 2.2. Assumptions and validity o the model A 1-3 composite is considered to be an equivalent layered composite Hull and Clyne 1996, Kar-Gupta and Venkatesh 27, Kumar and Chakraborty 29 where the second phase layer thickness is proportional to the iber volume raction. The equivalent series and parallel structure is shown in ig.3. The ollowing assumptions have been employed based on the undamental concept such as i the overall homogenized stress equilibrium condition in the transverse directions axes -1 and 2, ii rule o mixtures concept in the longitudinal direction or stresses axis-3, and iii the homogenized displacement conditions or the longitudinal case in the theoretical calculations Sakthivel and Arockiarajan 21: In this model, 1 the normal eective stresses along longitudinal direction axis - 3 and the shear stress in 1-2 plane are predicted based on the individual stress and volume raction constituents o the iber and the matrix. 2 The stresses along the transverse directions - 1&2 and the shear stresses in 1-3 & 2-3 planes are considered to be equal or the iber, the ceramic and the 1-3 composite, by considering the overall homogeneous stress equilibrium. 3 Concerning the normal eective strain along longitudinal direction and the shear strain in 1-2 plane are considered to be equal or the iber, the matrix and the 1-3 composite. 4 The eective strain along the transverse directions are the shear strains on the 1-3 & 2-3 planes are calculated based on the individual strain and volume raction constituents o the iber and the ceramic. 5 The piezocomposites are dielectric in nature the eective electric displacements along the longitudinal direction can be summed by averaging over the two phases based on a simple theory o series and parallel connectivity. However, the dielectric displacements are taken to be the same in the transverse directions or all. 6 The electric ield along the longitudinal direction is considered to be the same or iber and ceramic and 1-3 composite constituents. Along the other two directions, the electric ields are considered to be zero Eective parameter expressions o the 1-3 composite The ormulation starts with replacing the strain components and electric ield o matrix with the iber and composite strains and electric ield based on the assumptions. This calculation gives the strain components and electric ield to be only unctions o the iber and the composite. Subsequently, by equating the stress components along the longitudinal direction o ceramic and 1-3 composite, we can get a relation or the iber strain components in terms o composite strain as well as electric ield. Substitution o the derived resulting equation into eq. 6 will thus give the eective electromechanical coeicients o the composite as

6 6 International Journal o Structural Changes in Solids C33 c = V C 31 A1 + C 32 B1 + C 33 + C 35 D1 e 31 E1 C31A m 1 C32B m 1 + C 33 m 1 V κ c 33 = V e 31 A2 + e 32 B2 + κ 33 + e 35 D2 κ 31 E2 e m 31A 2 e m 32B 2 + κ m 33 1 V e c 33 = V e 31 A1 + e 32 B1 + e 33 + e 35 D1 κ 31 E1 e m 31A 1 e m 32B 1 + e m 33 1 V β33 c = V C 31 A3 + C 32 B3 + β 33 + C 35 D3 e 31 E3 C31A m 3 C32B m 3 + β 33 m 1 V P3 c = V e 31 A3 + e 32 B3 + P 3 + e 35 D3 κ 31 E3 e m 31A 3 e m 32B 3 + P 3 m 1 V 7 where the required constant A i, B i,d i and E i can be reerred rom the Appendix -A. Eq. 7 shows the eective parameter expressions o 1-3 composite with 9 degree poling direction o matrix phase class II material. Similarly, eq. 8 shows the the eective properties or the class V material wherein the iber rod is rotated with some angle θ rom the longitudinal axis. C33 c = V C 31 S3 + C 32 K3 + C 33 Cm 31S 3 + C32K m 3 e m 32F V C 33 e c 33 = V e 31 S3 + e 31 K3 + e 33 em 35M 3 κ c 33 = V e 31 S6 + e 31 K6 + κ 33 em 35M V κ m 33 β33 c = V C 13 S7 + C 23 K7 + β 33 Cm 13S 7 + C23K m 7 e m 13F V 8 β33 m P3 c = V e 31 S7 + e 31 K7 e m 31S 7 e m 31K 7 + P V P3 m where the required constant S i, K i,and M i can be reerred rom the Appendix -B. The perormance evaluation o piezoelectric composites or various applications has been discussed in the literature; reer Kar-Gupta and Venkatesh 27, Della and Shu 28. Figure o merit help assess the utility o a structure or particular applications. Three important igures o merit - the piezoelectric charge coeicient d h, the acoustic impedance Z, and the coupling constant K t are typically invoked to characterize the utility o piezocomposite in practical applications. 3. Results and discussion The eective properties o the proposed 1-3 piezocomposite material or dierent iber and matrix poling directions are calculated using the model derived in the previous section. In the present work, the iberrod is chosen as TLZ-5, and the matrix is VDF/TrFE copolymer that possess high value o piezoelectric and pyroelectric coeicients when compared to other polymer materials. Table 1 shows the material parameters o the ceramic iber and copolymer matrix that are used in the calculations. In the calculations, and m are reerred to the parameters o TLZ-5 material and VDF/TrFE copolymer parameters. This work ocuses on capturing the eect o ceramic rod volume ractions on mechanical, electrical and thermal characteristics o the 1-3 piezocomposite materials. The results are discussed into two parts, namely: eective properties o 1-3 composite due to matrix orientation and variation o composite properties due to iber orientation. In order to validate the proposed model, the simulated results are compared with other approaches. Fig. 4 let shows the variation o stiness constant C 11 with iber volume raction. The simulated results are compared with the numerical approach Srihari 211 in which the interace eects are considered and also compared with multiscale approach proposed by Nan et al. 2. The graphs show that the model is in good agreement with the other approaches or most o the values and the percentage deviation lies between 1 15%. Similarly, Fig. 4 right shows the variation o acoustic impedance Z with iber volume raction and compared with other approaches as well as experimental data rom literature Taunaumang et al The graphs are in comparable agreement with the experimental data and the percentage o variation is upto 2%. Such deviations may be attributed due to the grain boundry eects and the distribution o iber. In this model, the interace eects are not considered. However, urther research will be ocussed in this context.

7 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites 7 Table 1: Parameters o the iber and matrix used in the model calculations Taunaumang et al Parameter TLZ VDF/TrFE C11 E GP a C12 E GP a C13 E GP a C33 E GP a C33 D GP a C44 E GP a C66 E GP a e 15 C/m e 31 C/m e 33 C/m κ ε 11, κ ε 22 C/V m 1.55 x x 1 11 κ ε 33 C/V m 1.65 x x 1 11 α 11, α 22, α 33 C x x 1 6 P 1, P 2, P 3 C/K/m x x 1 5 ρ kg/m Eective thermo-electro-mechanical properties Matrix rotation The rods are aligned along the direction - 3. For class I type composites, the iber and the matrix are poled along the same direction. However, the composites that belong to classes II and IV 9 and 18, the material properties o matrix have been rotated appropriately. In the subsequent discussions, the results o class IV materials are not presented, since the class II and IV materials have an identical results. In general, the sensitivity o transducers are measured based on the electro-mechanical coeicient reers to energy conversion between mechanical to electrical and vice-versa. It is desired to have larger coupling constants being more desirable. The operating bandwidth o the devices can be determined by using the electromechanical coupling constant K t. The variation o eective electromechanical coupling constant K t or the 1-3 type piezocomposite with respect to the volume raction o iber rods V or dierent poling direction o matrix class I-IV is shown in ig. 5. The electro-mechanical coeicient increases gradually as volume raction o iber increases due to the act that the electrical energy is more conveniently converted into the mechanical energy. As the high end o iber volume raction approaches, the K t value reduces towards the value o iber due to lateral clamping o the iber by the polymer. At the medium range o ceramic rod volume raction.2 < V 1 <.8, K t value is almost a constant value. It is desired to have larger coupling constants being more desirable or the design aspects. In this context, it shows that the maximum K t value can be achieved between 4 to 8 % iber volume raction. K t value is unaected by the dierent matrix poling directions. The acoustic impedance Z is an important parameter or the determination o acoustic transmission and relection at the boundary o two materials having dierent acoustic impedances. In order to enhance the perormance o the transducer, good impedance matching between the transducer and the surrounding media e.g., water, human tissue is desired. The bulk piezoceramic materials typically have higher acoustic impedance than the surrounding environment. For instance, the acoustic impedance o a piezoceramic transducer is around 3 MRayl, while the surrounding environment blood or sot tissue is in the order o 1.5 MRayl. The impedance parameter can be signiicantly enhanced by appropriate design o a composite structure, that renders the development o piezoelectric polymer materials such as PVDF opened up the new possibilities or transducers operating at low acoustic impedance. The acoustic impedance o piezocomposite materials can vary in the order o 1 MRayl. Fig. 6 demonstrates that combination o good acoustic properties can be achieved by making optimal selection o ceramic rod volume raction. It also shows that the acoustic impedance increases linearly as the volume raction o iber increases except or some deviations at the high iber volume raction. For the dierent kind

8 8 International Journal o Structural Changes in Solids Multiple scattering approach Current approach Numerical approach Multiple scattering approach Current approach Numerical approach Experimental results[tau,94] c C GPa ZMRayl Fiber volume ractionv Fiber volume ractionv Figure 4: let Variation o stiness constant C 11 with the ceramic rod volume raction V ; Right Variation o acoustic impedance Z with the ceramic rod volume raction V Srihari 211 o poling directions o matrix class I- IV, the impedance properties do not have much variations. The essential requirement or designing a transducer is the low acoustic impedance and the high electro-mechanical coupling coeicient. From the designer prospective; the requirement o impedance or underwater and bio-medical imaging applications are in the order o 2 to 12 MRayl. This requirement can be achieved by 1 to 2 % iber volume raction but dierent matrix poling direction do not have any impact on the impedance values. The present study shows that 1-3 composites can be superior in both aspects such as minimizing impedance and maximizing coupling actor. The eectiveness o piezocomposite or the hydrophone applications can be represented by the hydrostatic piezoelectric coeicient, and hydrostatic voltage coeicient. d h captures the eective strength o the electromechanical coupling in a piezoelectric material, esp., in the conversion o hydrostatic mechanical loads to electrical signals. In applications such as in hydrophones, large values o the piezoelectric charge co-eicient is desirable in order to achieve sensitivity to the detection o sound waves. Fig. 7 shows the eective hydrostatic charge coeicient d h o the piezoelectric composite with respect to the volume raction o the ceramic rod. The pattern o curve shows that the charge coeicient o 1-3 type composite decreases as the ceramic rod volume raction V increases over 3%. It is also observed that this coeicient is very high at low ceramic rod volume raction or 9 degree poling direction compared with, and 18 degree poling directions. From design consideration, the maximum d h can be achieved around 3% o iber volume raction with, and 18 degree poling directions. With 9 matrix poling direction, there is a steep increase and decrease behavior and may not it or optimized design. The design o 1-3 piezocomposite or underwater application is ocused on the hydrostatic pressure response igure o merit FOM that needs to be maximized d h g h by varying the matrix and iber material properties, and iber volume raction. Owing to the improvements in their d h and g h values, the 13 type samples had high FOM values. The eective hydrophone igure o merit FOM is plotted as a unction o ceramic rod volume ractions in ig. 8, wherein this value does not deviate much or Class I and III materials. However, FOM varies signiicantly when the poling o matrix is at 9. Since FOM is direct product o d h and g h, the maximum FOM can be achieved around 3% o iber volume raction with, and 18 degree poling directions. With 9, and 27 matrix poling direction, there is a steep increase and decrease behavior which might not sutiable or tansducer applications. The piezocomposites are used as transducer in the active structures which are subjected to thermal environments. Hence, the transducer material should have high heat resistance, strength and stiness in order to withstand the thermal impacts. When the 1-3 piezoelectric materials are exposed to thermal loads, they exhibits pyroelectric eect and generates temporary electric charges due to the thermal strain. This eect can also be used or applications like inrared detector, pyrometer, and energy harvesting, etc. A new pyro-electric material

9 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites K t Volume raction o iber V Figure 5: Variation o electromechanical coupling constant K t with the ceramic rod volume raction V Z MRayls Volume raction o iber V Figure 6: Variation o acoustic impedance Z with the ceramic rod volume raction V d h pc/n Volume raction o iber V Figure 7: Variation o hydrostatic charge constant d h with the ceramic rod volume raction V

10 1 International Journal o Structural Changes in Solids x d h g h 1-15 m 2 /N Volume raction o iber V Figure 8: Variation o hydrostatic igure o merit d h g h with the ceramic rod volume raction V 4 x 1-5 Pyroelectric coeicient C/K/m Volume raction o iber V Figure 9: Variation o pyroelectric coeicient P 3 with the ceramic rod volume raction V 3 25 Piezoelectric coeicient C/m Volume raction o iber V Figure 1: Variation o piezoelectric coeicient e 33 with the ceramic rod volume raction V

11 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites 11 can be tailored by combining the bulk ceramic and polymer phase with appropriate proportions and properties. The plot shows the non-linear variation o pyroelectric coeicient in longitudinal direction P 3 or class I and III type materials Fig. 9. The pyroelectric coeicient decreases up to the ceramic rod volume raction o around 8% due to the domination o the matrix coeicient over the iber value. The piezoelectric coupling coeicient is directly proportional to the electrical to acoustic conversion that determines the transducer eiciency. Linear variation is noticed in Fig. 1 or most o the volume raction o the iber but there is a minimal decrease in the value at high iber volume raction Eective thermo-electro-mechanical properties Fiber rotation This section concentrates on analyzing the eective properties o the 1-3 composite with respect to the dierent iber poling direction. Fig. 11 shows the non-linear variation o the electromechanical coupling coeicient K t or various iber poling directions as a unction o the iber volume raction. As the orientation o the iber rods deviates rom the 3 - axis, i.e. θ >, there is a signiicant reduction in the K t value irrespective o iber volume raction. It is desired to have larger coupling constants being more desirable or the design aspects. However, dierent iber poling direction gives reduction in the K t values. Hence, it is not desirable to use dierent iber orientation or transducer design. The acoustic impedance Z shown in Fig. 12 is a useul parameter or biomedical imaging application and the predicted results shows that the acoustic impedance increases linearly as the volume raction o the iber increases except or some deviations at the high iber volume raction. The variations o acoustic impedance between the dierent iber poling directions are minimal. From the designer prospective; the requirement o impedance or underwater and bio-medical imaging applications are in the order o 2 to 12 MRayl. Fig. 13 shows the eective hydrostatic charge coeicient d h o the piezoelectric composite with respect to the volume raction o the ceramic rod. The pattern o curve shows that the charge coeicient o 1-3 type composite decreases as the ceramic rod volume raction V 1 increases over 35% or the poling angle θ =. However, the orientation o the iber rods deviates rom the 3 - axis, i.e. θ >, this value decreases at high iber volume raction. The eective hydrophone igure o merit FOM as a unction o ceramic rod volume ractions is shown in ig. 14. The variation in FOM is direct consequence o d h and g h since it is the direct product o d h and g h. FOM is direct product o d h and g h and the maximum FOM can be achieved around 9% o iber volume raction with 3 degree poling directions. With 6, and 9 iber poling direction, FOM eect is negligible and might not sutiable or tansducer applications. The plot shows the non-linear variation o pyroelectric coeicient in longitudinal direction P 3 or dierent iber poling directions Fig. 15. The pyroelectric coeicient decreases up to the ceramic rod volume raction o around 8% due to the domination o the matrix coeicient value over the iber value except perpedicular iber poling direction θ = 9. I the poling direction o iber deviates rom θ = 9 then at high iber volume raction this constant decreases. The coupling coeicient e 33 varies linearly with volume raction over most o the range Fig. 16. With dierent iber poling direction, the e 33 value is almost same Optimal design problem This section begin with a description o the design parameters or transducer design using 1-3 piezocomposite. The ollowing parameters are considered as: I Volume raction o the PZT rods: All o the properties o the composite are very sensitive to the volume raction o the PZT rods and hence it is one o the main design variables o the problem. II Arrangement o the PZT rods: Piezoelectric composite materials are transversely isotropic in nature. Given the properties and volume ractions o the PZT rods and the matrix, the hydrophone characteristics are uniquely deined by the eective transverse bulk modulus that depends on the spatial arrangement o the PZT rods. Hence the hydrophone perormance characteristics are determined by the inluence o the shape and distribution o the rod. III Properties o the matrix: The hydrophone characteristics are very sensitive to the properties o the matrix material. Avellaneda and Swart 1994 showed that introducing additional porosity into the isotropic polymer matrix may dramatically improve the perormance o the composite. It is also reported that

12 12 International Journal o Structural Changes in Solids K t Volume raction o iber V Figure 11: Variation o electromechanical coupling constant K t with the ceramic rod volume raction V Z MRayl Volume raction o iber V Figure 12: Variation o acoustic impedance Z with the ceramic rod volume raction V d h pcn Volume raction o iber V Figure 13: Variation o hydrostatic charge constant d h with the ceramic rod volume raction V

13 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites x d h g h X1 15 m 2 /N Volume raction o iber V Figure 14: Variation o hydrostatic igure o merit d h g h with the ceramic rod volume raction V 2.6 x Pyroelectric coeicient C/K/m Volume raction o iber V Figure 15: Variation o pyroelectric coeicient P 3 with the ceramic rod volume raction V 3 Piezoelectric coeicient C/m Volume raction o iber V Figure 16: Variation o piezoelectric coeicient e 33 with the ceramic rod volume raction V

14 14 International Journal o Structural Changes in Solids decreasing the Poisson s ratio o the matrix may result in enhanced perormance o the composite. It is important to explore the idea o optimally designing the matrix or the piezoelectric composite. IV Properties o the piezoceramic ibers: The moduli, dielectric constant o the piezoceramic rod is very important that should combine with the polymer on order to get the enhanced properties. V Fiber poling directions: Due to manuacturing diiculties, the iber rod alignment may not be parallel to the loading direction in the 1-3 piezocomposites. This manuacturing issue can be considered or optimal design. VI Matrix poling directions: Since the piezoelectric coeicients o the ceramic and polymer are o opposite signs and signiicantly dierent, it may be possible to pole the two phases in opposite directions, that will give enhanced coupling coeicient due to the add-on contribution rom active iber as well as matrix. This idea can be incorporated or design considerations. Having deined the optimal design parameters, the eective transducer hydrophone and bio-medical application perormance coeicients can be maximized based on the ollowing unctionals: 1 The electromechanical coupling constant K t 2 The acoustic impedance Z 3 The longitudinal velocity V D 33 4 The eective hydrostatic charge coeicient d h 5 The eective hydrostatic voltage coeicient g h 6 A common hydrostatic igure o merit - FOM d h g h 7 The eective electromechanical coupling actor k h In this work, the above mentioned optimal parameters and the eective transducer perormance unctionals are being calculated using the layered approach reer sec. 2.3 and the simulated results are discussed in detail in section 3. Additionally the predicted perormance unctional variations or dierent iber and matrix poling direction are tabulated; see Tables. 2 and 3. Based on the previous discussion, the perormance coeicients can be predicted in terms o the volume raction o the piezoelectric rods and the iber, matrix poling directions. Based on the problem in hand, we can optimize the possible combination to get the enhanced properties or transducer application. Consider a case study wherein the requirement is to maximize the hydrostatic charge coeicient d h, the hydrostatic voltage coeicient g h and the Figure o Merit FOM. The iber and matrix material properties are used rom the table 1 or this analysis and the optimum solution can be achieved or 2 to 3 % o iber voulme raction. Concerning the dierent poling directions, it is appreciable to use and 18 matrix poling directions. The other possible matrix and iber poling directions cannot it the requirment due to the signiicant variations with respect to iber volume raction. From this study, the design procedure is suggested as ollows: 1 Take a iber and matrix material, described by a individual properties and an equivalent layered model; 2 Find the eective material properties or piezocomposites based on the iber volume raction using the proposed layered approach; 3 Find the eective piezocomposite properties as unctions iber and matrix poling directions; 4 Find the hydrophone and bio-medical perormance coeicients; 5 Find optimal volume raction V by perorming the optimization procedure until convergence; 6 Perorm sensitivity analysis with respect to iber volume raction; 7 Change volume raction and poling directions using optimization technique; 8 Go to step 2 repeat until convergence.

15 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites 15 Table 2: Variations o perormance parameters or dierent matrix poling direction [1ex] V Angle K t Z V D d h g h FOM 1 3 k h P β MRayl m/s pc/n mv.m/n p/p a C/K/m 2 N/m 2 / C

16 16 International Journal o Structural Changes in Solids Table 3: Variations o perormance parameters or dierent iber poling direction V Angle K t Z V D d h g h FOM 1 3 k h P β MRayl m/s pc/n mv.m/n p/p a C/K/m 2 N/m 2 / C

17 Sakthivel and Arockiarajan / Eective characteristics o 1-3 piezoelectric composites Summary In this work, an analytical method based on parallel and series theory has been devised or 1-3 piezoelectric composites. The perormance o these composites as transducers or underwater and biomedical applications has been studied. The model is used to calculate the complete eective behavior o the piezoelectric composite subjected to combined thermo-electro-mechanical loading conditions. The eects o variations in the material properties under dierent iber and matrix poling directions were studied. It is observed that the variations in the elastic, piezoelectric and dielectric material constants with ceramic rod volume raction are nonlinear. The analytical study shows that an appropriate selection o the poling characteristics o the individual iber rod and matrix phases could lead to the development o a piezocomposite or speciic devices. The signiicant variations on eective igures o merit FOM were evaluated or ceramic rod volume raction and poling characteristics on the perormance o active 1-3 piezocomposites in bio-medical and underwater applications. 5. Appendix - A A 1 = 1 V A C13 m C 13 + B C23 m C 23 C 35 C De 13 A 2 = 1 V A e m 13 e 13 B e m 23 e 23 Ce 35 Dκ 13 A 3 = 1 V A β11 m β 11 B β22 m β 22 + D P1 m P 1 B 1 = 1 V B C13 m C 13 + E C23 m C 23 C 35 F Ge 13 B 2 = 1 V B e m 13 e 13 E e m 23 e 23 F e 35 Gκ 13 B 3 = 1 V B β11 m β 11 E β22 m β 22 + G P1 m P 1 D 1 = 1 V C C13 m C 13 + F C23 m C 23 C 35 K Me 13 D 2 = 1 V C e m 13 e 13 F e m 23 e 23 + Ke 35 + Mκ 13 D 3 = 1 V C β11 m β 11 F β22 m β 22 + M P1 m P 1 E 1 = 1 V D C13 m C 13 G C23 m C 23 + C 35 M Ne 13 E 2 = 1 V D e m 13 e 13 + G e m 23 e 23 + Me 35 + Nκ 13 E 3 = 1 V D β11 m β 11 + G β22 m β 22 + N P1 m P 1 [ 1 Q 1 = V κ 22 + ] [ V κ m 22 1 V κ 22 + ] [ 1 V κ m 22 + V e 24 + ] 2 V e24 m [Q 1]{1 V A = C 11 +V C11} m B = C = D = E = F = K = G = M = N = [1 V C 22 +V C m 22][1 V C 55 +V C m 55][1 V κ 11 +V κ m 11] [Q 1 ]{1 V C 12 +V C m 12} [1 V C 22 +V C22][1 V m C 55 +V C55][1 V m κ 11 +V κ m 11] [Q 1 ]{1 V C15} [1 V C 22 +V C22][1 V m C 55 +V C55][1 V m κ 11 +V κ m 11] [ Q 1 ]{1 V e 11} [1 V C 22 +V C22][1 V m C 55 +V C55][1 V m κ 11 +V κ m 11] [Q 1 ]{1 V C 55 +V C m 55} [1 V C 22 +V C m 22][1 V C 55 +V C m 55][1 V κ 11 +V κ m 11] [ Q 1 ]{1 V e 15 +V e m 15} [1 V C 22 +V C22][1 V m C 55 +V C55][1 V m κ 11 +V κ m 11] [ Q 1 ]{1 V e 33} [1 V C 22 +V C22][1 V m C 55 +V C55][1 V m κ 11 +V κ m 11] [ Q 1]{1 V e 24 +V e m 24} [1 V C 22 +V C m 22][1 V C 55 +V C m 55][1 V κ 11 +V κ m 11] [Q 1]{1 V κ 11 +V κ m 11} [1 V C 22 +V C m 22][1 V C 55 +V C m 55][1 V κ 11 +V κ m 11] [Q 1]{1 V κ 33 +V κ m 33} [1 V C 22 +V C m 22][1 V C 55 +V C m 55][1 V κ 11 +V κ m 11] 9

18 18 International Journal o Structural Changes in Solids Appendix - B S 3 = A C13 m C 13 + B C23 m C 23 + C e m 13 K 3 = B C13 m C 13 + J C23 m C 23 D e m 13 M 3 = K C13 m C 13 + M C23 m C 23 P e m 13 S 6 = A e 31 + B e 32 K e m 35 K 6 = B e 31 + J e 32 M e m 35 F 3 = C 1 1 V C13 m C 13 + D 1 1 V C23 m C 23 + H 1 1 V e m 13 F 7 = C 1 1 V β11 m β 11 + D 1 1 V β22 m β 22 + H 1 1 V P1 m P 1 M 6 = K e 31 + M e 32 N e m 35 S 7 = A β11 m β 11 + B β22 m β 22 + C P1 m P 1 K 7 = B β11 m β 11 + J β22 m β 22 D P1 m P 1 A 11 = 1 V C 44 + V C44 m 1 V κ 22 + V κ m V e 24 1 V C 55 + V C55 m A = 1 V C 55 + V C55 m 1 V C 11 + V C11 m 1 V κ 11 + V κ m 11 1 V C 22 + V C22 m [1 V A 1 = A11 e V C 22 +V C22+1 V m C 55 +V C55V m e m 12 2] A B 1 = A 11 [1 V e V C 12 +V C12+1 V m C 55 +V C55V m e m 12 2] A C 1 = A 11 [1 V e V κ 22 +V κ m 22+1 V C 55 +V C55V m e m 12 2] A [1 V D 1 = A11 e V κ 22 +V κ m 22+1 V C 11 +V C11V m e m 12 2] A J 1 = A 11 [1 V e V κ 22 +V κ m 22+1 V e 24 +V e m 24V e m 11 2] A K 1 = A 11 [1 V C V κ 22 +V κ m 22+1 V C 55 +V C55V m e m 12 2] A [1 V M 1 = A11 C V κ 22 +V κ m 22+1 V C 11 +V C11V m C12 m 2] A N 1 = A 11 [1 V C V κ 22 +V κ m 22+1 V e 12 +V e m 12V e m 11 2] A H 1 = A 11 [1 V C V κ 22 +V κ m 22+1 V e 12 +V e m 12V e m 11 2] A [1 V P 1 = A11 C V κ 22 +V κ m 22+1 V e 24 +V e m 24V C12 m 2] A 1 Reerences Avellaneda, M. and Swart, P Calculating the perormance o 1-3 piezocomposites or hydrophone applications: an eective medium approach. J. Acoust. Soc. Am. 13, pp Avellaneda, M. and Swart, P Calculating the perormance o 1-3 piezocomposites or hydrophone applications:an eective medium approach, J. Acoust. Soc. Am. 13, pp Barnett, D. and Lothe, J Dislocations and line charges in anisotropic piezoelectric insulators, Phys. Status Solid-B 67, pp

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