Electrical Boundary Conditions Research Articles

Geometrical nonlinear free vibration responses of FG-CNT reinforced composite annular sector plates integrated with piezoelectric layers

The objective of this research is to scrutinize large amplitude vibration response of functionally graded carbon nanotube reinforced composite (FG-CNTRC) annular sector plates with surface-bonded piezoelectric layers. A nonlinear formulation is derived based on the first-order shear deformation theory (FSDT), von Karman geometrical nonlinearity along with the Hamilton principle. The distribution of electric potential through the thickness of the piezoelectric layers is simulated by a sinusoidal function. The closed circuit electrical boundary condition is taken into consideration for the top and bottom surfaces of the piezoelectric layers. The nonlinear dynamic equations, boundary conditions and Maxwell equation are discretized using the generalized differential quadrature method and direct iterative method is then employed to solve the nonlinear system of equations. The variation of nonlinear frequency versus the vibration amplitude is highlighted considering various influential parameters such as distribution and volume fraction of the CNTs, geometrical parameters, boundary conditions and the thickness of the piezoelectric layers. It is found that the dynamic responses of the CNTRC sector plate may be noticeably enhanced by adjusting values of the CNT volume fraction and distribution. The numerical results reveal that the increase in the nonlinear frequency descents at certain vibration amplitude owing to vibration mode redistribution.

Optimal switch timing for piezoelectric-based semi-active vibration reduction techniques

Piezoelectric-based semi-active vibration reduction techniques typically rely on rapid changes in the electrical boundary conditions or corresponding stiffness state. Approaches such as state switching and synchronized switch damping on a resistor or an inductor require four switching events per vibration cycle, with switch timing associated with displacement extrema. Any deviation from this switch timing affects the performance of these techniques. Typical harmonic forcing analyses focus on the energy dissipation and only evaluate the performance at resonance. This study evaluates displacement reduction for harmonic excitation, both at resonance and for frequencies near resonance. Furthermore, it examines the effect of sub-optimal switch timings. Numerical simulations of a non-dimensional model are performed, and an analytical solution is derived for any switch time. This analysis shows that the optimal switch timing depends on the forcing frequency relative to the natural frequency of the structure. Thus, the classical switch time at peak displacement is only optimal when the excitation is exactly at resonance. Even when the optimal switch timing is known, uncertainties in vibration sensing cannot guarantee that switches will occur at the desired moment. Therefore, this work characterizes the degradation in vibration reduction performance when switching away from the optimal switch time based on global, non-dimensional parameters.

Extended displacement discontinuity method for an interface crack in a three-dimensional transversely isotropic piezothermoelastic bi-material. Part 2: Numerical method

The extended displacement discontinuity method is proposed to analyze the nonlinear electric and thermal effects on an interface crack in a three dimensional (3D), transversely isotropic, piezothermoelastic bi-material under combined mechanical-thermo-electrical loadings. The fundamental solutions for uniformly distributed, extended displacement discontinuities applied over a triangular element are obtained by integrating the fundamental solutions for the unit-point, extended displacement discontinuities given by Part 1 over the triangular area. In order to eliminate the oscillatory singularity of the stresses near the crack front, the Delta function in the fundamental solutions is approximated by the Gaussian distribution function, and accordingly, the Heaviside step function is replaced by the Error function. The extended, displacement discontinuity boundary element method with an iterative approach is proposed to determine the value of the fields in the crack interior for opening-crack model. As an example, an elliptical interface crack is studied under different electrical and thermal boundary conditions. The extended stress intensity factors (SIFs) without oscillatory singularities, the forms of the energy release rate (ERR) and local J-integral which can be both expressed in terms of intensity factors are obtained. The numerical method is validated and the influence of combined loadings and material-mismatch on the results is studied. The effects of different boundary conditions and ellipticity ratio on the results are also investigated.

The constituent equations of piezoelectric cantilevered three-layer actuators with various external loads and geometry

This paper presents test results for deformation conditions of three-layer, piezoelectric cantilever converters subjected to various electrical and mechanical boundary conditions. A general solution has been developed based on implementation of piezoelectric triple segments (PTS) to the beam. A working mechanism and conditions for strain of the PTS segment have been determined. Basing on the general solution, for the cantilever actuator subjected to an external force (of single and dual PTS segments) and a uniform load (of single PTS segment), particular solutions have also been developed. Moreover, dimensionless frequencies of the oscillating motion for the analyzed converters have been determined. In the next step, the influence of such factors as length, quantity and position of PTS segments, their relative stiffness and unit weight on values of the obtained frequencies of vibration have been defined. The resulting analytical solutions have been compared with the developed FEM solution.

Acoustic Wave Sensors for Liquid Environments

Acoustic wave devices are theoretically investigated in their potential role of technological platform for the development of sensors for liquid environments. In this study, we theoretically studied the propagation of in-plane polarized Lamb modes, Love modes, Shear Horizontal Plate modes along layered structures, aimed at the design of acoustic waveguides suitable to trade-off mass sensitivity and low loss. The phase velocity shift and attenuation due to the presence of a viscous liquid contacting the device surface and to a mass anchored to the sensor surface are calculated for different waveguide structures, confirming that the sensor behavior can be enhanced by a proper choice of the material parameters (i.e., material types, crystallographic orientation, electrical boundary conditions, and thicknesses) of the layered waveguide structure.

An Investigation of Modeling Error of EIT Reconstruction

Electrical Impedance Tomography is an imaging method which attempts to reveal the conductivity distribution of a domain based on the electrical boundary condition. For time difference EIT, the voltage difference at two time steps is employed for reconstruction. This is an ill-posed inverse problem, especially, it is non-linear. The currently available EIT devices are all based on linearized reconstruction algorithms. The linearized reconstruction employs a reconstruction matrix which is essentially a regularized pseudo inverse of the Jacobian matrix. This reconstruction matrix multiplying the voltage differences will provide a distribution of conductivity changes. However, the linearized reconstruction contains modelling error. In this paper, we study the modelling error caused by linearization based on the shunt model through simulations. Specifying a current injection pattern in simulation, at each time step a simulated voltage measurement can be calculated from Maxwell’s equations. The voltage difference between two time steps can be obtained. On the other hand, according to the assumption of linearized reconstruction, the voltage difference is assumed to be the Jacobian matrix multiplying the conductivity distribution changes. The discrepancy between these two voltage differences will be studied.

Finite field methods for the supercell modeling of charged insulator/electrolyte interfaces

Interactions between aligned dipoles in periodic model systems are often considered spurious and removed by partitioning the supercell in isolated slabs separated by vacuum layers. One popular and efficient screening approach is the so-called dipole correction. In this work, the authors present an alternative finite-field based method which retains all electrostatic interactions and dispenses with vacuum layers. Adapting the constant-$D$ approach developed by Stengel, Spaldin, and Vanderbilt for the study of ferroelectric nanocapacitors, the new scheme here is intended for molecular dynamics simulations of the interface between an insulator carrying a specifically absorbed surface charge and an ionic conductor (the electrolyte) at finite temperature. The authors show that the dipole correction scheme is equivalent to $D$=0 electric boundary conditions with or without vacuum layers. Thus, both global polarization $P$ of the heterogeneous system and corresponding average macroscopic electric field $E$ are finite. The authors then continue with the investigation of the variation of $P$ with $E$ or $D$ and validate expressions based on $P$ for computing the electric double-layer capacitance using atomistic simulations.

Reflection and transmission of plane wave in multilayered nonlocal magneto-electro-elastic plates immersed in liquid

Taking the nonlocal effect into account, we investigate the reflection and transmission of the plane wave propagating in three-dimensional multilayered magneto-electro-elastic (MEE) plates which are immersed in liquid. Based on the basic equations with nonlocal effect, the first-order state-space system is established to describe the relations of magnitudes of the extended stress and displacement. The general solutions of the extended stress and displacement are expressed in terms of the eigenvalues and eigenvectors which are derived from the first-order state-space system. Then the stiffness matrix method is employed to find the relation between the displacement and traction on the upper and lower interfaces of the layer. After defining the mechanical, electric and magnetic boundary conditions, we derive the reflection and transmission coefficients of the plane wave propagating in the MEE plates by deriving the global stiffness matrix. Finally, numerical examples are provided to show the effect of the nonlocal parameter, stacking sequences, frequency and incident angle on the reflection and transmission coefficients.

Nonlinear vibration analysis of piezoelectric plates reinforced with carbon nanotubes using DQM

The aim of the paper is to analyze nonlinear transverse vibration of an embedded piezoelectric plate reinforced with single walled carbon nanotubes (SWCNTs). The system in rested in a Pasternak foundation. The micro-electro-mechanical model is employed to calculate mechanical and electrical properties of nanocomposite. Using nonlinear strain-displacement relations and considering charge equation for coupling between electrical and mechanical fields, the motion equations are derived based on energy method and Hamilton\'s principle. These equations can\'t be solved analytically due to their nonlinear terms. Hence, differential quadrature method (DQM) is employed to solve the governing differential equations for the case when all four ends are clamped supported and free electrical boundary condition. The influences of the elastic medium, volume fraction and orientation angle of the SWCNTs reinforcement and aspect ratio are shown on frequency of structure. The results indicate that with increasing volume fraction of SWCNTs, the frequency increases. This study might be useful for the design and smart control of nano/micro devices such as MEMS and NEMS.

Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids

By constructing a new numerical conformal mapping and using the Stroh-type formulism, an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated. The explicit expressions of the complex potential function, field intensity factors, energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable. Numerical examples are presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions.

Influence of strain on an epitaxial ferroelectric (Ba0.50Sr0.50)TiO3 nanodot under different electrical boundary conditions

A first-principles-derived effective Hamiltonian approach is used to reveal the temperature-versus-misfit strain phase diagram of an epitaxial (Ba0.50Sr0.50)TiO3 dot under different electrical boundary conditions. The results indicate that the electrical polarization and toroidal moment are highly sensitive to the applied strain and/or electrical boundary conditions, resulting in a wide variety of phases that are not found in a free-standing BST dot and in bulk. The calculations indicate that within a narrow range of surface charge screening an intermediate phase in which the polarization and toroidal moment coexist. The dependences of the electrical polarization, toroidal moment and dielectric permittivity on the misfit strain and electrical boundary conditions at room temperature are also investigated and compared with the available theoretical predictions and experimental measurements.

A level-set based IGA formulation for topology optimization of flexoelectric materials

This paper presents a design methodology based on a combination of isogeometric analysis (IGA), level set and point wise density mapping techniques for topology optimization of piezoelectric/flexoelectric materials. The fourth order partial differential equations (PDEs) of flexoelectricity, which require at least C1 continuous approximations, are discretized using Non-Uniform Rational B-spline (NURBS). The point wise density mapping technique with consistent derivatives is directly used in the weak form of the governing equations. The boundary of the design domain is implicitly represented by a level set function. The accuracy of the IGA model is confirmed through numerical examples including a cantilever beam under a point load and a truncated pyramid under compression with different electrical boundary conditions. Finally, we provide numerical examples demonstrating the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization.

SH Waves in (1 –x)Pb(Mg1/3Nb2/3)O3- xPbTiO3 Piezoelectric Layered Structures Loaded with Viscous Liquid

The velocity dispersion and attenuation of shear horizontal (SH) waves in a layered piezoelectric structure loaded with viscous liquid is studied, where the (1 – x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 [PMN-xPT] single crystal is chosen as the piezoelectric layer. The PMN-xPT is being polarized along [011]c and [001]c so that the macroscopic symmetries are mm 2 and 4 mm, respectively. For the nonconductive liquid, the electrically open and shorted conditions at the interface between the liquid and the piezoelectric layer are considered. The phase velocity equations are derived analytically. The effects of the electrically boundary condition, the viscous coefficient and mass density of liquid as well as the thickness of the PMN-xPT layer on the phase velocity and attenuation are graphically illustrated. The results show that the phase velocity for the [011]c polarized PMN-0.29PT is much smaller than that for the [001]c polarized PMN-0.33PT, and the effects of viscous coefficient and piezoelectric layer thickness on the phase velocity for the [011]c case are stronger than that for the [001]c case. In addition, the electrical boundary conditions have an obvious influence on the propagation behaviors. These results can be useful for the designs and applications of acoustic wave devices and liquid biosensors.

Dielectric Breakdown Model for an Electrically Semi-Permeable Penny-Shaped Crack in Three-Dimensional Piezoelectric Media

The dielectric breakdown (DB) model for a penny-shaped crack under a semi-permeable boundary condition in a three-dimensional piezoelectric medium is studied. An approximate analytical solution is derived by using the boundary integral equation with extended displacement discontinuity, and the corresponding boundary element method with double iterative approaches is developed to analyze the semi-permeable crack. The effect of electric boundary conditions on crack faces is discussed on the basis of DB model. By comparing the DB model with the polarization saturation (PS) model for different piezoelectric materials, some interesting phenomena related to the electric yielding zone and local J-integral are observed.

Free vibration of functionally graded carbon nanotube reinforced composite plates integrated with piezoelectric layers

In the present research, free vibration behavior of carbon nanotube reinforced composite (CNTRC) plates integrated with piezoelectric layers at the bottom and top surfaces is analyzed. Plate is modeled according to the first order shear deformation plate theory. Distribution of CNTs across the plate thickness may be functionally graded (FG) or uniformly distributed (UD). Properties of the composite media are obtained according to a modified rule of mixtures approach which contains efficiency parameters. Distribution of electric potential across the piezoelectric thickness is assumed to be linear. The complete set of motion and Maxwell equations of the system are obtained according to the Ritz formulation suitable for arbitrary in-plane and out-of-plane boundary conditions. Besides, two types of electrical boundary conditions, namely, closed circuit and open circuit are considered for the free surfaces of the piezoelectric layers. Chebyshev polynomials are used as the basis functions in Ritz approximation. The resultant eigenvalue system is solved to obtain the frequencies of the system as well as the mode shapes. It is shown that, fundamental frequency of a closed circuit plate is always higher than a plate with open circuit boundary conditions.

Linear stability of buoyant convection in a horizontal layer of an electrically conducting fluid in moderate and high vertical magnetic field

Linear stability of buoyant convective flow in a horizontal layer of an electrically conducting fluid is considered with reference to horizontal Bridgman semiconductor crystal growth. The fluid flows owing to the horizontal temperature gradient in the presence of a vertical magnetic field. The main interest here is in the stability of the flow for a sufficiently strong magnetic field, for the Hartmann number Ha > 10, and increasing to high values, of the order of 103–104. The Prandtl number, Pr, has been fixed at Pr = 0.015. It is shown that besides the Hartmann number the instability strongly depends on the type of the thermal boundary conditions at the horizontal walls. For thermally conducting walls the basic temperature profile exhibits zones of unstable thermal stratification, which leads to instabilities owing to the Rayleigh-Bénard mechanism. However, the transitions between various, most unstable modes as Ha increases are not trivial. For sufficiently high values of Ha, the most unstable mode consists of transverse oscillatory rolls located in the region of unstable stratification. For thermally insulating walls, the transitions are simpler, and for sufficiently high Ha, the most unstable mode consists of longitudinal, steady, three-dimensional mode which is concentrated in the Hartmann layers at the horizontal boundaries. This mode has a combined dynamic-thermal origin and is owed to a strong shear in the Hartmann layers. The electrical boundary conditions do not qualitatively affect the picture of transitions between modes for both thermally conducting and thermally insulating walls.

Regularization of EIT reconstruction based on multi-scales wavelet transforms

Abstract Electrical Impedance Tomography (EIT) intends to obtain the conductivity distribution of a domain from the electrical boundary conditions. This is an ill-posed inverse problem usually solved on finite element meshes. Wavelet transforms are widely used for medical image reconstruction. However, because of the irregular form of the finite element meshes, the canonical wavelet transforms is impossible to perform on meshes. In this article, we present a framework that combines multi-scales wavelet transforms and finite element meshes by viewing meshes as undirected graphs and applying spectral graph wavelet transform on the meshes.

Transition to turbulence in Hunt's flow in a moderate magnetic field

Pressure-driven magnetohydrodynamic duct flow in a transverse uniform magnetic field is studied by direct numerical simulation. The electric boundary conditions correspond to Hunt's flow with perfectly insulating walls parallel to the magnetic field (sidewalls) and perfectly conducting walls perpendicular to the magnetic field (Hartmann walls). The velocity distribution exhibits strong jets at the sidewalls, which are susceptible to instability even at low Reynolds numbers Re. We explore the onset of time-dependent flow and transition to states with evolved turbulence for a moderate Hartmann number . At low Re time-dependence appears in the form of elongated Ting-Walker vortices at the sidewalls of the duct, which, upon increasing Re, develop into more complex structures with higher energy and then the sidewall jets partially detach from the walls. At high values of Re jet detachments disappear and the flow consists of two turbulent jets and nearly laminar core. It is also demonstrated that, there is a range of Re, where Hunt's flow exhibits a pronounced hysteresis behavior, so that different unsteady states can be observed for the same flow parameters. In this range multiple states may develop and co-exist, depending on the initial conditions.

Active vibration control of FGM plates with piezoelectric layers based on Reddy’s higher-order shear deformation theory

Literature researching the active vibration control of functionally graded material (FGM) plates with piezoelectric layers using Reddy’s higher-order shear deformation theory (HSDT) using any of the element-free methods does not exist. To the best of the authors’ knowledge, this paper is the first to use Reddy’s HSDT with the element-free IMLS-Ritz method to investigate this problem. In this study, seven mechanical degrees of freedom (DOF) and one additional electrical DOF are considered for each node of the discretized domain. The natural frequency results of two FGM plates with top and bottom piezoelectric layers are compared with the literature in terms of various electrical and mechanical boundary conditions, volume fraction exponent (n) and dimension ratios, with obvious agreement. Furthermore, parametric studies are performed, for the first time, to study the effects of mechanical boundary conditions, n value, FGM plate thickness-to-width ratio and piezoelectric layer thickness to FGM plate thickness ratio on the natural frequency increment between open and closed circuit conditions. For the purpose of active vibration control, a constant velocity feedback approach is utilized. The effectiveness of two proposed positions, of piezoelectric sensor and actuator layers, to control the vibration of FGM plates is investigated.

Untangling Electrostatic and Strain Effects on the Polarization of Ferroelectric Superlattices

The polarization of ferroelectric superlattices is determined by both electrical boundary conditions at the ferroelectric/paraelectric interfaces and lattice strain. The combined influence of both factors offers new opportunities to tune ferroelectricity. However, the experimental investigation of their individual impact has been elusive because of their complex interplay. Here, a simple growth strategy has permitted to disentangle both contributions by an independent control of strain in symmetric superlattices. It is found that fully strained short‐period superlattices display a large polarization whereas a pronounced reduction is observed for longer multilayer periods. This observation indicates that the electrostatic boundary mainly governs the ferroelectric properties of the multilayers whereas the effects of strain are relatively minor.