Isotropic Piezoelectric Materials Research Articles

Effect of quadratically varying electric potential on the multiple cracks in piezoelectric materials

A basic solution to the problem of three collinear equal cracks in a transversely isotropic piezoelectric material plate is obtained under the influence of electromechanical forces. Stroh formalism and complex variable methods are used to derive the mathematical expressions for SIF (stress intensity factor), COD (crack opening displacement), and COP (crack opening potential). The length of the saturation zones is investigated at each crack tip under the influence of quadratically varying electric displacement. Furthermore, a numerical study is carried out to discuss the effect of quadratically varying electrical potential on the saturation zone, COD, and COP. The results obtained numerically are reported graphically.

Precise solutions of dynamic problems in stratified transversely isotropic piezoelectric materials

Precise solutions of dynamic problems in stratified transversely isotropic piezoelectric materials

Fractal contact analysis for transversely isotropic piezoelectric materials: Theoretical and numerical predictions

This paper aims to investigate the micro frictionless and non-adhesive rough contact for transversely isotropic piezoelectric materials. The surface topography is formulated by the two-variable Weierstrass–Mandelbrot fractal function. The system of contact is subjected to light generalized loads. By virtue of the Hertzian solution of transversely isotropic piezoelectric materials and the size distribution function, the generalized loads-area relation is developed. The smallest truncated contact area in the theoretical model is determined by comparing the theoretical and simulation results. The present numerical solutions via the finite element method are validated in comparison to the numerical results in literature. The influence of various fractal parameters on the contact behavior of transversely isotropic piezoelectric solids are studied.

Analysis of reflection and transmission phenomenon at distinct bonding interfaces in a rotating pre-stressed functionally graded piezoelectric-orthotropic structure

The present mathematical study analyzes the reflection and transmission phenomenon of a plane wave being incident at the distinct types of separating interfaces in a rotating pre-stressed structure with two semi-infinite media comprised of functionally graded piezoelectric-orthotropic (FGPO) materials. The effects of pre-stresses, rotation, and functional gradients along with six different types of boundaries viz. Normal stiffness boundary (NSB), Transverse stiffness boundary (TSB), Electric imperfection boundary (EIB), Complete debonded boundary (CDB), Slip boundary (SB), and Welded contact boundary (WCB) on reflection coefficients (RCs) of reflected qP, qSV, and EA waves and transmission coefficients (TCs) of transmitted qP, qSV, and EA waves have been examined with a comparative approach. Comparison of the obtained result with that of the case when rotating pre-stressed structure with two semi-infinite media is constituted of lesser anisotropic functionally graded transversely isotropic piezoelectric (FGTIP) materials is also carried out. To validate the findings energy-partitions of reflected and transmitted waves for both the cases of the structures have been computed and shown to observe the Law of conservation of energy. Further, on considering a grazing incident of qP wave in both the cases of structure, RCs and TCs for reflected and transmitted qSV waves and EA waves become zero, whereas the RC for reflected and TC of transmitted qP wave gets out of phase. Some important peculiarities have also been highlighted through numerical results, which serve as a salient feature of the study.

Bending waves of a rectangular piezoelectric laminated beam

A simple nonlinear model is proposed in this paper to study the bending wave in a rectangular piezoelectric laminated beam of infinite length. Based on the constitutive relations for transversely isotropic piezoelectric materials and isotropic elastic materials, combined with some electric conditions, we derive the bending wave equation in a long rectangular piezoelectric laminated beam by using energy method. The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal beam direction. The shock-wave solution, solitary-wave solution and other exact solutions of the bending wave equation are obtained by the extended F-expansion method. And by using the reductive perturbation method we derive the nonlinear Schrodinger (NLS) equation, further more, the bright and dark solitons are obtained. For those soliton solutions, and some parameters derived by the process of solving soliton solutions, some conclusions are drawn by numerical analysis with some fixed conditions.

The plane strain analysis for one-dimensional hexagonal piezoelectric quasicrystals strip in aperiodical plan

A new stress potential function is introduced, the non periodic plane problem in one-dimensional hexagonal piezoelectric quasicrystals is discussed and the physical equation of the stress-strain relationship in the non periodic plane is constructed. The exact solution of the straight crack in the periodic direction of the one-dimensional hexagonal piezoelectric quasicrystal is obtained. As an application, the problem of straight crack perpendicular to the direction of quasi-periodical in one-dimensional hexagonal piezoelectric quasicrystal with long and narrow body is solved. When the width of the long body becomes infinitely large, the Griffith crack solution is obtained. The results show that the stress at the crack tip remains singularity, which is basically consistent with the crack problem that penetrates along the quasi periodic direction. When the phonon field and the phase field get to zero, the above analytical solution degenerates into the fracture problem of isotropic piezoelectric materials, the results are in agreement with the existing results.

Mathematical modeling of the electrostressed state in the orthotropic piezoelectric space with an arbitrary orientated circle crack under uniaxial tension

A mathematical model for the analysis of the stress state in an orthotropic electroelastic material with an arbitrary orientated circular crack is developed. The model is based on the consideration of the coupled system of equations of static electroelasticity. The problem on electric and stress states in orthotropic piezoelectric space with an arbitrary orientated circular crack under homogeneous loads was considered. The solution of the problem was obtained by means of the triple Fourier transform and Fourier image of Green's function for an infinite anisotropic piezoelectric medium. This approach was tested in the case of the location crack in the isotropy plane of transversely isotropic piezoelectric material for which there was an exact solution of the problem. The comparison of the calculated results confirmed the efficiency of the used approach. Numerical experiments were carried out and distributions of stress intensity factors along the front of the arbitrary orientated circular crack in orthotropic piezoelectric materials under the uniaxial tension were studied.

Asymmetric impermeable wedge crack in a piezoelectric material under anti-plane deformation

A problem of an inclined wedge crack in transverse isotropic piezoelectric material under electromechanical loading is analyzed. A single anti-plane mechanical and in-plane electrical loads are applied at a point either on wedge surface or on crack surface. By using Mellin transform, the problem based on the generalized displacement and stress vector is formulated. The derived equation is solved employing the Wiener-Hopf technique, from which the intensity factors and the energy release rates are obtained in closed forms. With material properties of PZT-5H, the intensity factors and energy release rate for wedge and crack surface loadings are calculated numerically and investigated. The closed form solutions can be used as a Green’s function for arbitrarily coupled anti-plane mechanical and inplane electrical loading cases.

The intrinsic piezoresponse in piezoelectric medium under contact-mode piezoresponse force microscopy

The rapid development of the piezoresponse force microscopy (PFM) makes the investigation of the electromechanical coupling phenomena at the nanoscale to become possible in various non-centrosymmetric polar materials. However, it still remains challenging in accurately and quantitatively determinate the intrinsic piezoresponses from the detected displacement responses in PFM experiments. Here, we develop the fully coupled theory to quantitatively analyze the piezoresponses in transversely isotropic piezoelectric materials under contact-mode measurements by using the Hankel transform method, considering the profiles of the conductive scanning probe microscopy (SPM) probe. Based on this fully coupled method, we investigate the electroelastic fields in piezoelectric materials. Furthermore, the effective piezoelectric coefficients are discussed under mechanical and electrical loadings. We find out that the effective piezoelectric coefficients are influenced by external loadings and geometries of SPM probe under contact-mode measurements. These results remind us that it should be extremely careful on the quantitative investigation of the piezoelectric materials via contact-mode measurements in PFM experiments.

Experiment determinant method for an internal strain and electric field state in transversely isotropic piezoelectric materials by the torsion examination

Experiment determinant method for an internal strain and electric field state in transversely isotropic piezoelectric materials by the torsion examination

Stress State of an Orthotropic Piezoelectric Material with an Elliptic Crack

The electric and stress state of an orthotropic electroelastic medium with an elliptic crack acted upon by mechanical and electric loads is analyzed. To solve the problem, the triple Fourier transform and Fourier-image of the Green function for an infinite orthotropic piezoelectric medium are used. The approach is tested by solving the problem for a crack located in the isotropy plane of a transversely isotropic piezoelectric material, for which the exact solution exists. The comparison of the results of calculations confirms the high efficiency of the approach. The distribution of stress intensity factors along the front of the crack in an orthotropic electroelastic material under uniform loading is studied numerically

Modeling of Elastic Waves in a Fluid Loaded and Immersed Piezoelectric Hollow Fiber

Modeling of elastic waves in an infinite fluid-loaded and immersed homogeneous, transversely isotropic piezoelectric circular fiber is studied using linear elasticity theory. The equations of motion of the fiber are formulated using the constitutive equations of a transversely isotropic piezoelectric material. The equations of motion of the internal and external fluids are formulated using the constitutive equations of an inviscid fluid. Displacement potentials are used to solve the equations of motion of the fiber and the fluids. The frequency equation of the coupled system consisting of the fiber and the internal and external fluid is developed under the assumption of no-slip boundary condition at the fluid–solid interface. The computed non-dimensional frequencies, non-dimensional wave number, phase velocity, attenuation and electro mechanical coupling are plotted in the form of dispersion curves for the lead zirconate titanate (PZT-4) ceramics.

Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeam based on strain gradient and surface stress elasticity theories under initial stress using differential quadrature method

ABSTRACTThe size-dependent effect on free vibration of double-bonded isotropic piezoelectric Timoshenko microbeams using strain gradient and surface stress elasticity theories under initial stress is presented. This article is developed for isotropic piezoelectric material. Due to the high surface-to-volume ratio, surface stress has an important role with micro- and nanoscale materials. Thus, the Gurtin–Murdoch continuum mechanic approach is used. Governing equations of motion are derived by Hamilton's principle and solved by the differential quadrature method. The effects of pre-stress load, surface residual stress, surface mass density, surface piezoelectrics, Young's modulus of surface layers, three material length scale parameters, thickness to material length scale parameter ratios, various boundary conditions, and two elastic foundation coefficients are investigated. It is concluded that the effect of pre-stress load in greater modes is negligible for higher aspect ratios and this effect is similar to lower aspect ratios. Also, the size-dependent effect on the dimensionless natural frequency for strain gradient theory is higher than that for modified couple stress theory and classical theory, which is due to increasing stiffness of the Timoshenko microbeam model. Moreover, the results show that dimensionless natural frequency affects more by considering the material length scale parameters with respect to surface effect. The results are compared with the obtained results from the literature and show good agreement between them. It is concluded that the amplitude of the transverse displacements (w0) for a microbeam (MB) is more than the transverse displacements (w1) for a piezoelectric microbeam (PMB). On the other hand, using a piezoelectric layer for PMB, the amplitude of the transverse displacements (w1) reduces considerably with respect to MB, in which this effect leads to increase the stiffness of the microbeam and stability of microstructures. With considering the piezoelectric layer, the obtained results can be used to control the amplitude and vibration of microstructures, prevent the resonance phenomenon, design smart structures, and can be employed for micro-electro-mechanical systems and nano-electro-mechanical systems.

Elastostatic solutions of a multilayered transversely isotropic piezoelectric system under axisymmetric loading

The precise integration algorithm (PIA) and the technique of dual vector formulation are extended to explore the axisymmetric response of multilayered transversely isotropic piezoelectric materials under the action of external forces. The planes of cross-anisotropy are assumed to be parallel to the horizontal surface. The mechanical and electrical loads distributed over a circular area can be applied either on the surface of the layered system or at the interface between neighboring layers with dissimilar materials. By means of the Hankel integral transformation and the approach of dual vector form, the governing partial differential equations are reduced to the standard ordinary differential matrix equation. The PIA is exploited to evaluate the first-order ordinary differential matrix equation in the transformed domain. As the PIA is a highly accurate algorithm, any desired accuracy of axisymmetric piezoelectric solutions can be accomplished. The whole calculation is based on standard matrix algebra, and computational effort can be saved to a great extent. Furthermore, dual vector forms of key equations can facilitate the combination of adjacent strata. Finally, several numerical examples are given to validate the accuracy of the proposed technique and to investigate the influence of loading positions, types of external forces, stratified characters, and weak and thin interlayer on the response of the layered piezoelectric medium.

Coupled electro-mechanical surface responses induced by electro-mechanical loading moving steadily on piezoelectric half-plane solids

This paper investigates the coupled electro-mechanical surface responses of piezoelectric materials under electro-mechanical surface loading moving steadily with subsonic speed on a piezoelectric half-plane boundary. The electro-mechanical surface responses are exhibited through the elements of four matrices L−1(v), S(v)L−1(v), Ω(v) and Γ(v) for general anisotropic piezoelectric materials. For monoclinic piezoelectric materials with the plane of symmetry at x3=0 and for transversely isotropic piezoelectric materials, explicit expressions of the elements of these four matrices are further exploited in terms of generalized elastic stiffnesses. Numerical example is given for PZT-5H material to illustrate the coupled phenomena of the mechanical (electrical) deformations induced by electrical (mechanical) loadings as the subsonic speed of electro-mechanical surface loading varies.

1103 保存積分による3次元ピエゾと等方性材料の接合体の特異応力場の強さの計算(OS8-1 材料力学と計算力学の新展開)

In the present study, the conservative integral formulation is developed for calculating the intensity of singularity at a vertex of the interface in three-dimensional transversely isotropic piezoelectric and isotropic material joints. External mechanical load is applied to the joints. In order to investigate the influence of integral area on the accuracy of the results, models with various integral areas are used. Finally, the distribution of stresses and electric displacements are compared with the FEM results.

Propagation of non-axisymmetric waves in an infinite soft electroactive hollow cylinder under uniform biasing fields

Based on Dorfmann and Ogden's nonlinear theory of electroelasticity and the associated linear incremental theory, the non-axisymmetric wave propagation in an infinite incompressible soft electroactive hollow cylinder under biasing fields is investigated. The biasing fields are uniform, including an axial pre-stretch and a radial stretch in the plane perpendicular to the axis of the cylinder as well as an axial electric displacement. Such biasing fields make the originally isotropic electroactive material behave during its incremental motion like a conventional transversely isotropic piezoelectric material, hence greatly facilitating the following analysis. The three-dimensional equations of wave motion in cylindrical coordinates are derived and exactly solved by introducing three displacement functions. The exact solution is expressed in terms of Bessel functions, and explicit frequency equations are presented in different cases. For a prototype nonlinear model of electroactive material, numerical results are given and discussed. It is found that the initial biasing fields as well as the geometrical parameters of the hollow cylinder have significant influences on the wave propagation characteristics.

The mathematical modeling for bulk acoustic wave propagation velocities in transversely isotropic piezoelectric materials

The objective of this paper is to study the bulk acoustic wave (BAW) propagation velocities in transversely isotropic piezoelectric materials, aluminum nitride, zinc oxide, cadmium sulfide and cadmium selenide. The bulk acoustic wave velocities are computed for each direction by solving the Christoffel’s equation based on the theory of acoustic waves in anisotropic solids exhibiting piezoelectricity. These values are calculated numerically and implemented on a computer by Bisection Method Iterations Technique (BMIT). The modification of the bulk acoustic wave velocities caused by the piezoelectric effect are graphically compared with the velocities in the corresponding non-piezoelectric materials. The results obtained in this study can be applied to signal processing, sound systems and wireless communication in addition to the improvement of surface acoustic wave (SAW) devices and military defense equipment.

Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

SummaryThe dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.

Analytical and semi-analytical modeling of effective moduli bounds: Application to transversely isotropic piezoelectric materials

In this article, analytical and semi-analytical models of upper and lower bounds for the effective moduli of transversely isotropic piezoelectric heterogeneous materials based on the generalized Hashin–Shtrikman variational principle are presented. Compact matrix formulations are used to derive closed-form bound expressions for coupled and uncoupled effective moduli. Analytical models are given for some uncoupled coefficients and simplified formulations for the others. For more narrow bounds, downstream and upstream bounds are developed based on an incremental procedure. Numerical predictions are performed based on the developed methodological approaches, and the obtained results showed the applicability and effectiveness of the proposed models for transversely isotropic elastic and piezoelectric composite materials with ellipsoidal reinforcements of different types and shapes.