Electro-elastic Coupling Research Articles

Electroactive differential growth and delayed instability in accelerated healing tissues

Guided by experiments contrasting electrically accelerated recovery with natural healing, this study formulates a model to investigate the importance of electroactive differential growth and morphological changes in tissue repair. It underscores the clinical potential of leveraging electroactive differential growth for improved healing outcomes. The study reveals that voltage stimulation significantly enhances the healing and growth of biological tissues, accelerating the regeneration process across various growth modalities and steering towards isotropic growth conditions that do not favor any specific growth pathways. Enhancing the electroelastic coupling parameters improves the efficacy of bioelectric devices, initiating contraction and fortification of biological tissues in alignment with the electric field. This process facilitates swift cell migration and proliferation, as well as oriented growth of tissue. In instances of strain stiffening at elevated strains, the extreme critical growth ratio aligns with the predictions of neo-Hookean models. Conversely, for tissues experiencing strain stiffening under moderate to very low strain conditions, the strain stiffening effect substantially delays the onset of electroelastic growth instability, ultimately producing a smooth, hyperelastic surface devoid of any unstable morphologies. Our investigation, grounded in nonlinear electroelastic field and perturbation theories, explores how electric fields influence differential growth and instability in biological tissues. We examine the interactions among dimensionless voltage, internal pressure, electroelastic coupling, radius ratio, and strain stiffening, revealing their effects on promoting growth and delaying instability. This framework offers insights into the mechanisms behind electroactive growth and its instabilities, contributing valuable knowledge to the tissue healing.

Nonlinear analysis on electro-elastic coupling properties in bended piezoelectric semiconductor beams with variable cross section

In this paper, the effects of nonlinear constitutive relation on the electro-elastic coupling behaviors in non-uniform piezoelectric semiconductor beams are investigated. On the basis of three-dimensional theory for piezoelectric semiconductor and double power series, one-dimensional bending model is developed. In order to deal with the nonlinear problem, a differential quadrature method based iteration procedure is proposed. Comparing with the results from finite element method, the proposed method has good convergence and high accuracy. In numerical analysis, the effects of nonlinearity and non-uniformity in piezoelectric semiconductor are discussed thoroughly. It is found the introduction of nonlinearity induces the loss of symmetric for all electrical field quantities but little changes the mechanical quantities. While the reduction of width along the length direction produces larger mechanical and electrical responses. Importantly, the effects of nonlinearity on the electrical field quantities are more evident in a piezoelectric semiconductor beam with rapidly varied cross section. Besides, we designed the piezoelectric semiconductor beams subject to double shear forces and a beam with locally varied cross section. In the designed structures, the potential barriers and wells are realized, which provide new approaches adjusting the electro-elastic behaviors in piezoelectric semiconductors. The study in this paper could be the guidance designing high performance electronic devices.

The adaptive hygrothermo–magneto–electro–elastic coupling improved enriched finite element method for functionally graded magneto–electro–elastic structures

In this study, the hygrothermo–magneto–electro–elastic coupling improved enriched finite element method (HC-IEFEM) is proposed to analyze functionally graded magneto–electro–elastic (FG-MEE) structures. The static behavior of FG-MEE structures in the hygrothermal environment is studied. In the enriched finite element method (EFEM) enhanced by interpolation cover functions, the improved shape functions are added to solve the rank defect problem. Quadrilateral elements are used in the analysis of FG-MEE structures in this study. This improved method has proven to be relatively more efficient and resistant to mesh distortion than the traditional finite element method (FEM). Besides, the adaptive mesh refinement (AMR) scheme is used to refine the mesh in local areas to improve numerical calculation efficiency. Numerical examples show that the adaptive HC-IEFEM can achieve relatively high accuracy for analyzing FG-MEE structures. The proposed HC-IEFEM can obtain high accuracy in the hygrothermal environment using a relatively coarse mesh through the improved shape function and AMR scheme. Therefore, the significant potential is demonstrated in analyzing FG-MEE structures in the hygrothermal environment using the proposed adaptive HC-IEFEM.

Structural distortion driven enhancement of exchange bias effect in Cu1−xCoxCr2O4

We investigate exchange bias (EB) effect in two members with x = 0.4 and 0.6 of multiferroic Cu1−xCoxCr2O4 series, which are close to the interface between two structures at room temperature and represent with structures having I41/amd and Fd3¯m space groups, respectively. Both the compounds exhibit EB effect having EB field (HE) of 244 and 870 Oe at 2 K for x = 0.4 and 0.6, respectively with a cooling field of 10 kOe, which decrease with increasing temperature and disappear near their ferrimagnetic ordering temperatures (TN) close to 100 K. We note that higher HE correlates the higher coercivity (HC), which is found highest for x = 0.6 in the entire series of compounds. Our low temperature synchrotron diffraction studies for x = 0.6 confirm magnetoelastic as well as electroelastic coupling observed close to TN and ferroelectric ordering temperature (TFE). A structural transition to a polar Ima2 structure from I41/amd is observed close to 154 K, which coincides with TFE. Below TFE, thermal variation of structural distortion defined by (a − b)/(a + b) demonstrates significant magnetoelastic coupling. More importantly, the distortion parameter is found significantly higher than the rest member of the series and provides a hint for the highest corecivity in the entire series and correlates the higher EB effect. Current study demonstrates a correlation between structural distortion and EB effect and provides a clue of improving EB effect through the crystal structural engineering.

On the dynamics and stability of size-dependent symmetric FGM plates with electro-elastic coupling using meshless local Petrov-Galerkin method

In present study, the equations of motion of sandwich symmetric functionally graded size-dependent plates integrated with piezoelectric layers are derived using Hamilton’s variational principle (with modified Lagrangian) based on a higher-order nonlocal strain gradient theory in conjunction with Reddy’s third-order shear deformation plate kinematics. For the first time, a set of sixth-order partial differential equations are derived and solved by meshless local Petrov-Galerkin method to investigate the bifurcation buckling and free vibration of plates under electromechanical in-plane forces with considered nanostructure and various boundary conditions. The shifted and scaled moving least-squares approach is utilized for approximation of unknown variables, and the Gauss weight function is used as test function for deriving local weak form of governing equations. The Gauss-Legendre quadrature is employed for numerical evaluation of weak forms. A power-law distribution and a half cosine variation are applied to model the variation of materials properties and electric potentials, respectively. The obtained results are verified with well-known solutions in the literature. The effect of nonlocal parameter, material length scale parameter, power-law index, predefined electric field, axial compressive and tensile forces, volume ratio of FG core and piezoelectric layers, and stiffness of Pasternak foundation on the critical buckling loads and natural frequencies are presented and discussed comprehensively. The results obtained can be used in the analysis, design and control of composite nanostructures that are widely used in MEMS and NEMS devices.

Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Piezoceramics Containing Regular N-Sided Holes

In this paper, the calculation method of dynamic stress concentration around piezoelectric ceramics containing regular n-sided holes under the action of electroelastic coupling wave was studied, and it was applied to promising barium calcium zirconate titanate material. First, electroelastic governing equations were decomposed by using the auxiliary function method, and the solution forms of the elastic wave field and electric field were obtained by using the wave function expansion method. Then, the triangular boundary was simplified to a circular boundary using the mapping function, and the corresponding modal coefficients were determined according to simplified boundary conditions. Finally, the dynamic stress-concentration factor was calculated to characterize the dynamic stress concentration. We performed numerical simulations with a correlation coefficient of (1 − x)[(Ba0.94Ca0.06) (Ti0.92Sn0.08)]-xSm2O3-0.06 mol% GeO2 (abbreviated as (1 − x)BCTS-xSm-0.06G). The numerical calculation results show that the incident wave number, piezoelectric properties, shape parameters of the hole, and deflection angle have a great influence on the dynamic stress around the defect, and some significant laws are summarized through analysis.

Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Triangular Defect Piezoceramics

In this paper, a method to calculate the dynamic stress concentration around the triangular defect of piezoelectric material under electroelastic coupling is studied and applied to the promising barium calcium zirconate titanate. Firstly, the electroelastic governing equation is decomposed by decoupling technique, and the analytical solutions of elastic wave field and electric field are obtained by wave function expansion method. Then, the conformal transformation is used to simplify the triangle boundary into a circular boundary, and the corresponding modal coefficients are determined according to the simplified boundary conditions. Finally, the analytical solution of the dynamic stress concentration factor can be obtained according to the constitutive equation. Substitute the relevant material parameters of (Ba0.85Ca0.15)(Zr0.1Ti0.9)O3 and set different temperatures, Ce doping amount, and incident wave number for numerical simulation. The numerical results show that the incident wave number, piezoelectric properties, and the shape parameters and deflection angle of the triangular defect have a great influence on the dynamic stress around the defect, and some meaningful laws are summarized through analysis.

Elastic wave localization and energy harvesting defined by piezoelectric patches on phononic crystal waveguide

The application of piezoelectric effect to localize elastic wave and collect energy has been a hot research field in recent years. In this paper, we designed a phononic crystal (PnC)-based system with a piezoelectric waveguide to realize wave localization. (PnC) is one representative example of a metamaterial, it is constituted by a periodic repetition of unit cells. The most important feature of PnC is to forbid wave propagate in some special frequency, people called it bandgap. In our research, we introduce line defect waveguide and piezoelectric waveguide in the bandgap. By adding piezoelectric patches in the waveguide of PnC structure, the frequency of waveguide mode can be raised from 2240 Hz to 2265 Hz. The frequency difference between the two waveguides helps us create a wave localization in the position of piezoelectric patches. We also find that the localization performs best when the frequency is 2257 Hz. Besides, the electro-elastic coupling is studied in this paper, the coupling would become stronger when the two piezoelectric patches get closer, then we can get a vibration peak in strong coupling, it can help us localize wave if we need a very strong vibration in a point or a small position. On the contrary, the coupling would become weaker when the two piezoelectric patches get farther, and then we can obtain a wide localization position with the same amplitude of vibration between two piezoelectric patches, it can help us localize wave if we need the same vibration in a relatively wide area. We also investigate the relationship between power and the resistance in the circuit, results show that the maximum power appears in 103Ω of all the three cases. The research of wave localization and energy harvesting can give us a new idea to design many vibration and energy-localization devices.

On the piezoelectric effect on stability of symmetric FGM porous nanobeams

In present paper, the piezoelectric effect on bifurcation buckling of symmetric FGM porous nanobeam is presented based on a higher-order nonlocal elasticity and strain gradient theory in conjunction with Reddy third-order shear deformation beam theory. The strain energy density and electric enthalpy density functions are used to derive constitutive relations for porous functionally graded, as well as piezoelectric materials. Equations of motion for elastically supported layered FGM nanobeam with diverse distribution of porosity and electro-elastic coupling are derived based on the modified Hamilton’s variational principle. The formulated boundary value problem is analytically solved. For the first time, the effects of porosity distribution, volume of voids, distance between porosity and FGM core surfaces as well as material gradation, layers size, aspect ratio, stiffness of Kerr foundation, and external electric voltage on critical in-plane force, critical porosity and critical voltage were comprehensively presented and discussed. The investigation enables analysis and precise multi-dimensional control of static response of smart beam-like nanostructures widely used in MEMS and NEMS devices.

Modeling and simulating the effects of a general imperfect interface in fibrous piezoelectric composites

Interfacial behavior within fibrous piezoelectric composites (PZCs) may dramatically degrade the overall electromechanical performance and simultaneously vary the effective coupling moduli of these intelligent materials. In the present paper, we first recapitulate a physics-based general interfacial relation, which is rigorously derived from an ideal three-phase configuration and thus owns explicit physical meaning, to characterize this behavior between two different piezoelectric media. Afterwards, the explicit strong and weak governing equations of fibrous PZC boundary value problems (BVPs) are constructed, and a computational approach is then developed with the help of the extended finite element method (XFEM) to account for the discontinuities within both the primary (i.e., the electric potential and displacement fields) and the secondary fields (i.e., the normal electric displacement and normal traction fields). To achieve a benchmark problem with analytical exact solution, a simplified general interfacial relation is introduced, and a fibrous PZC BVP involving a cylindrical simplified general interface is designated and analytically solved. Hereafter, the convergence performance and validity of the elaborated computational approach are tested in detail. Eventually, discussions are further made on the influence of material composition and inhomogeneity shapes on the electroelastic coupling behavior of PZCs, and a few concluding remarks are drawn.

Phononic band gap of a quarter-wave stack for enhanced piezoelectric energy harvesting

Extraordinary phenomenon (e.g., band gap) exhibited by a phononic crystal (PnC) has recently been incorporated into piezoelectric energy harvesting (PEH) as a means to amplify input waves for improving the power generation capability. For a systematic design of PnC-based PEH using band gap reflection, it is required to identify a standing wave pattern (i.e. location of nodes and antinodes) in order to determine the optimal placement of a piezoelectric layer. In this context, we propose a quarter-wave stack (QWS)-based PEH system under elastic waves. The foundational idea of this study is to manipulate a standing wave pattern formed in a host medium by considering the phase change within the QWS. The key to creating an optimal standing wave for PEH is to intentionally form a displacement node at the interface between the host medium (here, continuous copper bar) and QWS at a target design frequency; thereby, the placement of the piezoelectric layer can be systematically determined in order that its center is aligned with the strain antinode to generate the maximum output power. To deeply explore the effect of the standing wave pattern on the output performances of the QWS-based PEH system, two key parameters are thoroughly investigated: 1) the number of unit cells and 2) the excitation frequency. Numerical analysis results suggest the followings: First, an increase in the number of unit cells corresponds to the effect of amplifying input loading for enhanced PEH; thereby, the output performances increase gradually and converge asymptotically. Second, since the standing wave pattern varies with the excitation frequency, the electroelastic coupling varies accordingly. These new findings provide guidelines on design parameters that can be manipulated to yield the best performance of the QWS-based PEH system under elastic waves for various potential applications.

Biological cells and coupled electro-mechanical effects: The role of organelles, microtubules, and nonlocal contributions

Biological cells are exposed to a variety of mechanical loads throughout their life cycles that eventually play an important role in a wide range of cellular processes. The understanding of cell mechanics under the application of external stimuli is important for capturing the nuances of physiological and pathological events. Such critical knowledge will play an increasingly vital role in modern medical therapies such as tissue engineering and regenerative medicine, as well as in the development of new remedial treatments. At present, it is well known that the biological molecules exhibit piezoelectric properties that are of great interest for medical applications ranging from sensing to surgery. In the current study, a coupled electro-mechanical model of a biological cell has been developed to better understand the complex behaviour of biological cells subjected to piezoelectric and flexoelectric properties of their constituent organelles under the application of external forces. Importantly, a more accurate modelling paradigm has been presented to capture the nonlocal flexoelectric effect in addition to the linear piezoelectric effect based on the finite element method. Major cellular organelles considered in the developed computational model of the biological cell are the nucleus, mitochondria, microtubules, cell membrane and cytoplasm. The effects of variations in the applied forces on the intrinsic piezoelectric and flexoelectric contributions to the electro-elastic response have been systematically investigated along with accounting for the variation in the coupling coefficients. In addition, the effect of mechanical degradation of the cytoskeleton on the electro-elastic response has also been quantified. The present studies suggest that flexoelectricity could be a dominant electro-elastic coupling phenomenon, exhibiting electric fields that are four orders of magnitude higher than those generated by piezoelectric effects alone. Further, the output of the coupled electro-mechanical model is significantly dependent on the variation of flexoelectric coefficients. We have found that the mechanical degradation of the cytoskeleton results in the enhancement of both the piezo and flexoelectric responses associated with electro-mechanical coupling. In general, our study provides a framework for more accurate quantification of the mechanical/electrical transduction within the biological cells that can be critical for capturing the complex mechanisms at cellular length scales.

Geometry Independent Direct and Converse Flexoelectric Effects in Functionally Graded Dielectrics: An Isogeometric Analysis

In the present study, a functionally graded material (FGM) based dielectric composite is investigated for direct and converse flexoelectric effects induced due to material inhomogeneity. An Isogeometric analysis based formulation is employed to effectively capture the gradient terms appearing in the electro-elastic coupling constitutive laws of flexoelectricity. Non-uniform stresses and strains generated due to non-uniform elastic properties in the domain induce flexoelectricity inherently without any special geometric and/or boundary conditions. A two-dimensional (2D) FGM structure made by the combination of polyvinylidene fluoride (PVDF) and barium titanate (BaTiO3) is studied under mechanical and thermal loading conditions for its flexoelectric response. The induced polarization and voltage along with its dependence on grading index (n) is analyzed. Further, to extend the study to converse flexoelectricity, the mechanical response of the FG flexoelectric material under an applied electric potential is studied. Finally, the actuation of the structure by an applied electric field is studied. As high as 10% actuation of original displacement is observed for higher values of n. Effectiveness of the grading concept to induce flexoelectric response is established through numerical studies. Longitudinal and transverse modes of flexoelectricity are scrutinized and the concept of grading is proved to inherently induce the flexoelectricity in dielectrics without special geometric modifications.

Frequency band gaps in dielectric granular metamaterials modulated by electric field

Wave propagation in granular materials is known to be dispersive. Micromorphic continuum model based upon granular micromechanics (Misra, A. and P. Poorsolhjouy, Continuum Mech. Thermodyn, 2016. 28(1-2): p. 215-234.) has the ability to describe this dispersion behavior. In this paper we show that the dispersive behavior can be modulated by using electric field when the grains have dielectric properties. To this end, we apply the recently enhanced model that incorporates electro-elastic coupling by connecting microstrain to electric dipole and quadrupole densities due to bound charges in dielectric grains (Romeo, M., Mech. Res. Commun., 2018. 91: p. 33-38.). We particularly investigate the effect of induced polarization that arises due to an imposed electric field. Two cases of dielectric one dimensional infinite rods with the same micromorphic properties have been studied, where case 1 and 2 are in null and nonzero external electric fields, respectively. Parametric studies are performed to understand the contribution of the polarizability (dipole effect), intrinsic quadrupole density, and external electric field on the dispersive behavior of granular media. Results predict an acoustic and an optical branch in the dispersive curve. Polarizability and external electric field are mainly affecting small wavenumber behavior of the wave branches, while quadrupole density alters the behavior of the material at large wavenumbers. A possibility of altering the optical branch to an acoustic branch is also observed, for which instability or attenuation occurs depending upon the direction of the imposed electric field with respect to the wave propagation direction. We find that the location and the width of the frequency band gaps can be altered using external electric field. The possibility of creating or removing frequency band gaps is also shown to exist. The extended theory accounting for electro-elasticity can therefore be utilized as a tool to analyze existing granular media, or to design granular metamaterials, as it systematizes the design process and eliminates ad-hoc manners leading to large data libraries.

A moving thermal dielectric crack in piezoelectric ceramics with a shearing force applied on its surface

• A dynamic, thermal dielectric crack model is proposed for piezoelectric materials. • Explicit expressions for the thermal, electric and elastic quantities are given. • The electric displacement inside the opening crack is determined. • Thermal stress intensity factor is presented for real or complex eigenvalues. • Influence of the crack moving velocity is shown. This article conducts an exact analysis of a thermal dielectric crack moving in piezoelectric materials. Self-generating thermal and electric loadings by the crack interior are exerted on the crack surfaces as well as various external loadings including a shearing force. Fundamental solutions of the thermal and electro-elastic coupling fields are given by determining a temperature function and a harmonic function with eigenvalues properties due to material properties considered. Analytical expressions are obtained benefiting evaluation of key parameters. Numerical analysis is done and some interesting observations are found. There is a critical crack velocity within and beyond which the electric loading exerts different influences on the thermal flux of crack interior and the thermal stress intensity factor.

Nonlinear electroelasticity: material properties, continuum theory and applications.

In the last few years, it has been recognized that the large deformation capacity of elastomeric materials that are sensitive to electric fields can be harnessed for use in transducer devices such as actuators and sensors. This has led to the reassessment of the mathematical theory that is needed for the description of the electromechanical (in particular, electroelastic) interactions for purposes of material characterization and prediction. After a review of the key experiments concerned with determining the nature of the electromechanical interactions and a discussion of the range of applications to devices, we provide a short account of the history of developments in the nonlinear theory. This is followed by a succinct modern treatment of electroelastic theory, including the governing equations and constitutive laws needed for both material characterization and the analysis of general electroelastic coupling problems. For illustration, the theory is then applied to two simple representative boundary-value problems that are relevant to the geometries of activation devices; in particular, (a) a rectangular plate and (b) a circular cylindrical tube, in each case with compliant electrodes on the major surfaces and a potential difference between them. In (a), an electric field is generated normal to the major surfaces and in (b), a radial electric field is present. This is followed by a short section in which other problems addressed on the basis of the general theory are described briefly.

Dynamic effective property of fibrous piezoelectric composites with spring- or membrane-type imperfect interfaces

The present paper studies the dynamic effective property of piezoelectric composites embedded with cylindrical piezoelectric fibers under anti-plane harmonic electro-elastic waves. By using the dynamic generalized self-consistent method (DGSM) of electro-elastic coupling wave, the problem of randomly distributed cylindrical fibers in a piezoelectric medium can be analyzed in terms of a representative volume element with a coated fiber embedded in an equivalent effective medium. The interfaces between the fibers and the matrix are assumed to be imperfect which are here modeled as spring- or membrane-type interfaces. Through wave function expansion method and an iterative method, the effective piezoelectrically stiffened shear modulus and the effective wave number are obtained. Examples are conducted to verify the present solutions and to illustrate the dependence of the effective piezoelectrically stiffened shear modulus on the wave number (frequency) as well as the interface properties. The special size effect related to interfacial imperfection is also discussed.

Piezoelectric metacomposite structure carrying nonlinear multilevel interleaved-interconnected switched electronic networks

In this article, a new class of piezoelectric phononic meta-composite structure carrying nonlinear multilevel interleaved-interconnected switched electronic networks is proposed, and its electro-elastic coupling between elastic waves and electrical waves is investigated. Normally, distributed multimodal resonant damping can be used to achieve several separate resonant-type band gaps for elastic wave propagation control in electromechanical phononic metamaterials. In the proposed approach, several resonant-type band gaps are induced by the peaks and valleys of electrical waves. Inductances adopted in the electronic network are besides much smaller than those required for generating resonant-type band gaps through conventional resonant circuit shunts. Such feature of requiring less or no inductance contributes to the generation of low-frequency band gaps in a semi-passive way. Numerical results show that the number of band gaps over the primitive pass band (between adjacent primitive stop bands) is determined by the level of the proposed nonlinear interleaved-interconnected electronic network. Compared with broadband wave propagation control, the proposed method is more flexible, and can easily target several discrete frequency domains of interest without significantly affecting other frequency domains. Through the theoretical and experimental investigation on low-frequency damping performance, wave control capability and damping performance are confirmed in the low frequency domain.

Coupling of experimentally validated electroelastic dynamics and mixing rules formulation for macro-fiber composite piezoelectric structures

Piezoelectric structures have been used in a variety of applications ranging from vibration control and sensing to morphing and energy harvesting. In order to employ the effective 33-mode of piezoelectricity, interdigitated electrodes have been used in the design of macro-fiber composites which employ piezoelectric fibers with rectangular cross section. In this article, we present an investigation of the two-way electroelastic coupling (in the sense of direct and converse piezoelectric effects) in bimorph cantilevers that employ interdigitated electrodes for 33-mode operation. A distributed-parameter electroelastic modeling framework is developed for the elastodynamic scenarios of piezoelectric power generation and dynamic actuation. Mixing rules (i.e. rule of mixtures) formulation is employed to evaluate the equivalent and homogenized properties of macro-fiber composite structures. The electroelastic and dielectric properties of a representative volume element (piezoelectric fiber and epoxy matrix) between two neighboring interdigitated electrodes are then coupled with the global electro-elastodynamics based on the Euler–Bernoulli kinematics accounting for two-way electromechanical coupling. Various macro-fiber composite bimorph cantilevers with different widths are tested for resonant dynamic actuation and power generation with resistive shunt damping. Excellent agreement is reported between the measured electroelastic frequency response and predictions of the analytical framework that bridges the continuum electro-elastodynamics and mixing rules formulation.

Interface energy on effective elastic constant of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain

ABSTRACTThe surfaces/interfaces effect is vital in nanocomposites. The electro-elastic surface/interface theory is introduced to predict the size-dependent effective elastic constants of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain, and the analytical solution is obtained. New boundary conditions governing the surface piezoelectricity are used to analyze the surface piezoelectricity effect. The average electro-elastic coupling field of the randomly distributed nano-particles with surface/interface effect is derived by using the effective field method. In the numerical examples, the interface effect under different material constituents is analyzed. It is found that the interface effect on the effective elastic constants is significantly related to the material properties of nano-particles. The elastic constants and piezoelectric constants show different effects on the effective elastic constants.