Piezoelectric Quasicrystals Research Articles

Surface Effects Study: A Continuum Approach From Fundamental Modes to Higher Modes and Topological Polarization in Orthotropic Piezoelectric Materials

Abstract The primary goal of the current work is to investigate how wave propagation influences the performance of surface acoustics wave (SAW) macro- and nano-sensors. Therefore, shear horizontal (SH) waves use the surface piezoelectricity theory to explore SH waves in an orthotropic piezoelectric quasicrystal (PQC) layer overlying an elastic framework (Model I), a piezoelectric substrate, and an orthotropic PQC substrate (model II). This study employs a variable-separable technique. The theoretical forms are constructed and used to present the wavenumber of surface waves in any direction of the piezoelectric medium, based on the differential equations and matrix formulation. In addition, we take into account the surface elasticity theory in order to obtain the phase velocity equation. Two configurations are examined: an orthotropic piezoelectric material layer over an elastic framework and a piezoelectric material half-space with a nanosubstrate. Analytical expressions for frequency equations are derived for both symmetric and antisymmetric waves. This study investigates the effects of surface elastic constants, surface density, anisotropic piezoelectric constant, and symmetric and antisymmetric modes on phase velocity. This study is confined to only linear wave propagation. Additionally, the analysis is based on idealized material properties, surface properties, and characteristic length of the material.

The interaction between a piezoelectric screw dislocation and collinear rigid lines in one-dimensional hexagonal quasicrystals

The interaction between a piezoelectric screw dislocation and collinear rigid lines in one-dimensional (1D) hexagonal quasicrystals (QCs) under anti-plane mechanical and in-plane electrical loading is analysed. Two idealized electrical assumptions, namely electrically conductive and dielectric, are considered. By using Muskhelishvilid’s complex transformation method, the explicit expressions for the field variables, the singularity of the field variables at the line tip and the force on the dislocation are obtained for a single rigid line. Based on the generalized Peach–Koehler formula, the generalized stress intensity factors and image force on dislocations in piezoelectric quasicrystal (PQC) are obtained. It is evident that the existence of phason field has an effect on image force.

Three-Dimensional Axisymmetric Analysis of Annular One-Dimensional Hexagonal Piezoelectric Quasicrystal Actuator/Sensor with Different Configurations

The presented article is about the axisymmetric deformation of an annular one-dimensional hexagonal piezoelectric quasicrystal actuator/sensor with different configurations, analyzed by the three-dimensional theory of piezoelectricity coupled with phonon and phason fields. The state space method is utilized to recast the basic equations of one-dimensional hexagonal piezoelectric quasicrystals into the transfer matrix form, and the state space equations of a laminated annular piezoelectric quasicrystal actuator/sensor are obtained. By virtue of the finite Hankel transform, the ordinary differential equations with constant coefficients for an annular quasicrystal actuator/sensor with a generalized elastic simple support boundary condition are derived. Subsequently, the propagator matrix method and inverse Hankel transform are used together to achieve the exact axisymmetric solution for the annular one-dimensional hexagonal piezoelectric quasicrystal actuator/sensor. Numerical illustrations are presented to investigate the influences of the thickness-to-span ratio on a single-layer annular piezoelectric quasicrystal actuator/sensor subjected to different top surface loads, and the effect of material parameters is also presented. Afterward, the present model is applied to compare the performance of different piezoelectric quasicrystal actuator/sensor configurations: the quasicrystal multilayer, quasicrystal unimorph, and quasicrystal bimorph.

Impact of frictional heat and electric charge on the thermomechanical contact properties of piezoelectric quasicrystals: A DC-FFT algorithm

Piezoelectric quasicrystals (PEQCs) are quasicrystals with piezoelectricity. It combines the excellent properties of piezoelectric materials and quasicrystal materials, and is expected to be used as actuators in fields such as aerospace, automotive, and intelligent manufacturing. This article investigates the physical and mechanical properties of one-dimensional hexagonal piezoelectric quasicrystals (1D HPEQCs) coating under normal force and frictional heat and current density by using the discrete convolution-fast Fourier transform (DC-FFT) algorithm-based hybrid element method (HEM). The general solution of the model in the two-dimensional Fourier domain is obtained by combining the two-dimensional Fourier transform with strict operator theory and the Almansi’s theorem. The DC-FFT algorithm-based HEM is used to efficiently and accurately cope with the issue of transforming the fundamental solution in the Fourier domain into the spatial domain. Therefore, the numerical solutions and distribution of stress and displacement in the phonon and phason fields are obtained. The results indicate that current density has a significant impact on the phason field, and an increase in current density enhances the mechanical behavior of the phason displacement. The research results can provide theoretical guidance for 1D HPEQCs as intelligent coating materials.

Three-Dimensional and Two-Dimensional Green Tensors of Piezoelectric Quasicrystals

Three-Dimensional and Two-Dimensional Green Tensors of Piezoelectric Quasicrystals

Analytical solutions to Mode I penny-shaped crack problems in two-dimensional hexagonal quasicrystals with piezoelectric effect

The paper studies a penny-shaped crack in an infinite three-dimensional body of two-dimensional hexagonal quasicrystal media with piezoelectric effect. The crack surfaces are applied combined electric and normal phonon loadings. Such a Model I crack problem is transformed into a mixed boundary value problem in the upper half-space, which is analytically solved using Fabrikant's potential theory method. The boundary integral-differential equations governing Model I crack problems are presented for two-dimensional hexagonal piezoelectric quasicrystals. The normal phonon displacement discontinuity and electric potential discontinuity across crack surfaces are taken as the unknown variables of boundary governing equations. Analytical solutions of all field variables are derived not only for the crack plane but also for the full space. Solutions in integral form are provided for the penny-shaped crack under arbitrarily distributed electric and normal phonon loadings. Closed-form solutions in terms of elementary functions are given for concentrated point loadings and uniformly distributed loadings, respectively. Key fracture mechanics parameters, such as crack surface extended displacements (i.e., normal phonon displacement, electric potential), crack tip extended stresses (i.e., normal phonon stress, electric displacement) distribution, and corresponding extended stress intensity factors, are clearly derived. Numerical results are utilized to verify the present analytical solutions and graphically illustrate the distribution of phonon-phason-electric coupling fields around the crack. The present solution can serve as a benchmark for both experimental and numerical investigations.

Mode-III fracture of four cracks emanating from an elliptical hole in one-dimensional orthorhombic quasicrystals with piezoelectric effect

In this study, a new generalized conformal mapping is introduced using the generalized complex variable method, focusing on the investigation of the anti-plane problem of an elliptical hole with four cracks in one-dimensional (1D) orthogonal piezoelectric quasicrystals. Under the consideration of electrically impermeable and electrically permeable boundary conditions, the governing equations of the anti-plane problem in 1D orthogonal piezoelectric quasicrystals are successfully simplified into an eighth-order partial differential equation and a fourth-order partial differential equation by introducing a potential function. Based on this, analytical solutions for the stress field and electric field are obtained, and the exact solution for the stress intensity factor (SIF) is derived. In addition, formulas for the electric displacement intensity factor (EDIF) and energy release rate are provided. Subsequently, the influence of geometric parameters and external loads on the field intensity factors and energy release rate is analyzed through numerical examples, providing a theoretical basis for engineering mechanics analysis.

Hamiltonian System for Three-Dimensional Problem of Two-Dimensional Decagonal Piezoelectric Quasicrystals and Its Symplectic Analytical Solutions

Hamiltonian System for Three-Dimensional Problem of Two-Dimensional Decagonal Piezoelectric Quasicrystals and Its Symplectic Analytical Solutions

Phase-field fracture modelling of piezoelectric quasicrystals

In this paper, an electro-elastic phase field model is developed for brittle fracture in piezoelectric quasicrystals. The elastic field of quasicrystals is composite of a phonon field and a phason field. The phonon field, phason field and electric field are coupled with each other, and the phonon and electric loadings trigger the variation of both the phonon and phason elastic energies, which then together contribute to crack evolution. The cracks initiation and propagation in piezoelectric quasicrystals under quasi-static mechanical loading and electric loading can be well predicted. Several numerical examples are performed to evaluate the effects of electric field and anisotropic fracture toughness. Simulation results show that the electric field influences the initial force, peak force, fracture displacement and crack path. For the single-edge notched tensile test, the effects of electric field force on the initial force, peak force and fracture displacement are obvious. For the single-edge notched shear test, the electric field force has no effect on the initial force, but it affects the peak force, fracture displacement and crack path. For the porous plate tensile test, a high-intensity electric field can deflect crack substantially resulting in passing through different holes. Further simulation indicates that the anisotropic fracture toughness has a significant effect on peak force, fracture displacement, and crack path. The present model can be used to simulate various fracture problems of piezoelectric quasicrystals.

Electromechanical coupling characteristics of multilayered piezoelectric quasicrystal plates in an elastic medium

AbstractQuasicrystals (QCs) have attracted tremendous attention of researchers for their unusual properties. In this paper, an exact electric‐elastic solution of the simply supported and multilayered three‐dimensional (3D) cubic piezoelectric quasicrystal (PQC) nanoplate with the nonlocal effect is derived. Based on the basic elasticity equation of 3D QCs, we construct the linear eigenvalue system in terms of the pseudo‐Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. The two‐parameter foundation model is utilized to simulate the interaction between the nanoplate and elastic medium. The propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. Based on the boundary conditions of the upper and lower surfaces of the laminate and foundation model, the solutions are employed to derive from the global propagator matrix. Compared with the conventional propagator matrix method, a new propagator method is reestablished to deal with numerical instabilities of the case of large aspect ratio and high‐order frequencies for QC laminates. Finally, typical numerical examples are presented to illustrate the influence of nonlocal parameters and elastic medium coefficients on phonon, phason, and electric variables of 3D PQC nanoplates.

Analysis of multilayered two-dimensional decagonal piezoelectric quasicrystal beams with mixed boundary conditions

Piezoelectric quasicrystals have a broad spectrum of promising applications and research value due to their unique atomic arrangement and multi-field coupling effects. This paper presents a mechanical analysis of the static bending and free vibration problems of two-dimensional multilayered decagonal piezoelectric quasicrystal laminated beams. Mechanical models of the beams are established, and the differential quadrature method in conjunction with the state-space method is utilized to obtain the solutions for the beams with mixed boundary conditions. We first investigated the static response and free vibration problem of quasicrystal laminated beams. By degrading them into crystalline materials, we verified the accuracy of the method by comparing the results with the existing ones. Then, the effects of different boundary conditions and slenderness ratios on the stresses, displacements, electric potentials, and electric displacements of quasicrystal laminated beams under normal mechanical loading and free vibration are investigated. Subsequently, the effects of different laying modes on the mechanical properties of the quasicrystal laminated beams when the material principal axis of the beam is changed are also analyzed.

On the Constitutive Modelling of Piezoelectric Quasicrystals

Quasicrystals endowed with piezoelectric properties belong nowadays to novel piezoelectric materials. In this work, the basic framework of generalized piezoelectricity theory of quasicrystals is investigated by providing an improvement of the existing constitutive modelling. It is shown, for the first time, that the tensor of phason piezoelectric moduli is fully asymmetric without any major or minor symmetry, which has important consequences on the constitutive relations as well as on its classification with respect to the crystal systems and Laue classes. The exploration of the tensor of phason piezoelectric moduli has a significant impact on the understanding of the piezoelectric properties of quasicrystals. Using the group representation theory, the classification of the tensor of phason piezoelectric moduli with respect to the crystal systems and Laue classes is given for one-dimensional quasicrystals. The number of independent components of the phason piezoelectric moduli is determined for all 31 point groups of one-dimensional quasicrystals. It is proven that the 10 centrosymmetric crystallographic point groups have no piezoelectric effects and that the remaining 21 non-centrosymmetric crystallographic point groups exhibit piezoelectric effects due to both phonon and phason fields. Moreover, the constitutive relations for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 9 with point group 6 and Laue class 10 with point group 6mm are explicitly derived, showing that the constitutive relations for piezoelectric quasicrystals depend on the considered Laue class as well as on the point group. Comparisons with existing results in the literature and discussion are also given.

Interaction of anti‐plane shear waves with two collinear cracks in 1D hexagonal piezoelectric quasicrystals

AbstractThe present article presents the interaction of anti‐plane shear waves by two collinear cracks in one dimensional (1D) hexagonal piezoelectric quasicrystals (PQCs). With the aid of Fourier transform, the mixed boundary value problem (MBVP) is transformed into three pairs of dual integral equations, which are solved analytically by Hilbert transform. The explicit expressions for dynamic stress intensity factors (DSIFs) of phonon and phason fields, crack opening displacement (COD) and electric displacement (ED) are derived in closed form and some special cases are studied. Numerical values of DSIFs of phonon and phason fields, COD and ED are plotted to show the effect of crack length and electric boundary condition. Moreover, the DSIFs of phonon field and COD are represented graphically for single crack.

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

This study examines the interference problem between screw dislocation and elliptical inclusion in one-dimensional hexagonal piezoelectric quasicrystals. The general solutions are obtained using the complex variable function method and the conformal transformation technique. When the screw dislocation is located outside or inside the elliptical inclusion, the perturbation method and Laurent series expansion are employed to derive explicit analytical expressions for the complex potentials in the elliptical inclusion and the matrix, respectively. Considering four types of far-field force and electric loading conditions, analytical solutions for various specific cases are obtained by using matrix operations. Expressions for the phonon field stress, phason field stress, and electric displacement are given for special cases, including the absence of a dislocation, the presence of an elliptical hole, and the interference between a screw dislocation and circular inclusion, as well as the case of a circular hole. The design and analysis of quasicrystal inclusion structures can benefit from the results of this work.

Dynamic Analysis of Multilayered Piezoelectric Quasicrystal Three-Dimensional Sector Plates with Imperfect Interfaces

Piezoelectric quasicrystals have attracted extensive attention due to their unique physical and mechanical properties. This paper studies the dynamic response of multilayered two-dimensional decagonal piezoelectric quasicrystal sector plates with imperfect interfaces. Based on the quasicrystal linear elasticity, partial differential state equations along the thickness direction are derived by using the state-space method. Then, by virtue of the differential quadrature method and the Fourier series expansions, this boundary-value problem with mixed boundary conditions and imperfect interfaces is solved. In addition, via the joint coupling matrix, the field quantities in the interior of the structure are connected to those on the external surfaces with numerical instability. Finally, parameter studies on the effects of angular spans, imperfect interfaces, and mixed boundary conditions are numerically investigated where the dimensionless frequencies and modes are exhibited.

An analytic solution of an arbitrary location through-crack emanating from a nano-circular hole in one-dimensional hexagonal piezoelectric quasicrystals

The mode III fracture performance of an arbitrary location through-crack at the edge of a nano-circular hole in one-dimensional hexagonal quasicrystals is studied. Based on the G-M elasticity theory, the piezoelectric quasicrystal theory and the boundary value problems theory, the phonon, phason and electric fields of the nano-defects (nano-hole and nano-cracks) are obtained, and the analytic expressions of the field intensity factors at both ends of through-crack are present. The present solution can be degenerated into the existing results. The relationship between the field intensity factors with the nano-defects related parameters, far-field loads and piezoelectric quasicrystal coupling coefficient are discussed. The field intensity factors have obvious size effects when the defects size is at nanoscale and the surface effect is considered. The field intensity factors tend to be classical theory with the increase in defect size. The field intensity factors show different changes with the increase in crack location angle and relative size between cracks. The size effect of the field intensity factors is significantly affected by the crack location. The far-field mechanical-electric loads and the piezoelectric quasicrystal coupling coefficient have obvious influence on the field intensity factors.

Study on effective electroelastic properties of one-dimensional hexagonal piezoelectric quasicrystal containing randomly oriented inclusions

In this paper, the effective electroelastic properties of one-dimensional (1D) hexagonal piezoelectric quasicrystal containing randomly oriented inclusions are considered. The explicit expressions are obtained for the Eshelby tensors for 1D hexagonal piezoelectric quasicrystals containing rod-shaped and penny-shaped inclusions. The closed forms of the electroelastic constants are acquired for four special cases of random orientations of inclusions. Numerical results are given for the 1D hexagonal piezoelectric quasicrystal containing randomly oriented ellipsoidal inclusions. The results indicate that the effective electroelastic properties of 1D hexagonal piezoelectric quasicrystal composites are strongly affected by both the aspect ratio and the orientation of inclusions.

Dynamic fracture of a partially permeable crack in a functionally graded one-dimensional hexagonal piezoelectric quasicrystal under a time-harmonic elastic SH-wave

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials.

Fracture mechanics of an arbitrary position crack emanating from a nano-hole in one-dimensional hexagonal piezoelectric quasicrystals

Fracture mechanics of an arbitrary position crack emanating from a nano-hole in one-dimensional hexagonal piezoelectric quasicrystals

Anti-plane problem of a nano-sharp crack in one-dimensional hexagonal piezoelectric quasicrystals with the electrically semi-permeable condition

Based on the Gurtin–Murdoch surface/interface theory and the complex potential theory, the problem of an electrically semi-permeable nano-sharp crack in one-dimensional (1D) hexagonal piezoelectric quasicrystals is analyzed. By constructing conformal mapping of the sharp crack, the analytic solutions of the electroelastic field are determined. Meanwhile, the field intensity factors and the energy release rate (ERR) under the electric boundary conditions of semi-permeable are obtained. Numerical examples are given to analyze the effects of the crack size, dielectric constant, electric loading, and mechanic loadings on the dimensionless field intensity factors and the dimensionless ERR. The results show that the impact of the surface effect becomes weaker along with the growth of the crack length, but increases with the increase in the crack angle. The change of the dielectric coefficient has little influence on the dimensionless stress intensity factors (SIFs), but has a strong influence on the dimensionless electric displacement intensity factor (EDIF). The obtained results are helpful to promote the development of nano-quasicrystal composite mechanics and provide an important theoretical support for the design and application of nano-QC materials.