One-dimensional Hexagonal Piezoelectric Quasicrystals Research Articles

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.

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.

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.

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.

Model III fracture analysis of a nanoscale elliptical hole with four cracks in one-dimensional hexagonal piezoelectric quasicrystals

When the defects are at the nanoscale, the failure characteristics of the piezoelectric quasicrystals are different from the macroscale case. The mode-III fracture problem of four nano-cracks emanating from a nanoscale elliptical hole in one-dimensional hexagonal piezoelectric quasicrystals was studied based on the Gurtin-Murdoch surface elasticity model. The analytical solutions of the electro-elastic field intensity factors and the energy release rate at the crack tip were obtained. The present solutions can be reduced to the existing results. The dimensionless electro-elastic field intensity factors show obvious size dependent effect. The influences of the defects size, the multi-cracks interaction, the phonon-phason field coupling coefficient and the applied electro-elastic loads on the fracture characteristics were discussed in detail.

Mode-III fracture of a nanoscale cracked hole in one-dimensional hexagonal piezoelectric quasicrystals

Mode-III fracture of a nanoscale cracked hole in one-dimensional hexagonal piezoelectric quasicrystals

Sliding frictional contact of one dimensional hexagonal piezoelectric quasicrystals coating on piezoelectric substrate with imperfect interface

This paper studied the sliding frictional contact of a one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) coating, where the PEQCs coating is imperfectly bonded to a transversely isotropic piezoelectric (PE) substrate. The frequency response functions (FRFs) of displacements and stresses of phonon (phason), electric potentials and electric displacements, are analytically derived by applying double Fourier integral transforms to the general solutions and boundary conditions of the 1D hexagonal PEQCs coating-PE substrate systems. The conjugate gradient method (CGM) is used to obtain the unknown pressure distribution and the discrete convolution-fast Fourier transform technique (DC-FFT) is involved to calculate the contact responses such as displacements and stresses. Numerical results are given to reveal the influences of coating thickness, loading conditions and imperfection indices on the contact behavior, which provide a reference basis for PEQCs coating experimental analysis and intelligent structures (systems) design.

Surface effect on two nanocracks emanating from an electrically semi-permeable regular 2n-polygon nanohole in one-dimensional hexagonal piezoelectric quasicrystals under anti-plane shear

In this paper, the conformal mapping from a regular 2[Formula: see text]-polygon hole with two collinear asymmetric cracks into a circle is constructed. Based on the Gurtin–Murdoch surface/ interface model and complex potential theory, two collinear asymmetric nanocracks emanating from an electrically semi-permeable regular 2[Formula: see text]-polygon nanohole embedded in an infinite one-dimensional hexagonal piezoelectric quasicrystals with surface effect are investigated. The size-dependent stress intensity factors of phonon field and phason field, electric displacement intensity factor at the nanocrack tip are derived for electrically semi-permeable boundary condition. Numerical examples are illustrated to show that the size of the hole, mechanical load, electric load, cracks relative size change with stress intensity factor of phonon field and electric displacement intensity factor. Also analyzed the change of the electric displacement intensity factor with different electric permeability at the nanocrack tip and the dimensionless intensity factor with [Formula: see text].

Frictional contact problem of one-dimensional hexagonal piezoelectric quasicrystals layer

Based on three-dimensional (3D) general solutions for one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs), this paper studied the frictional contact problem of 1D hexagonal PEQCs layer. The frequency response functions for 1D hexagonal PEQCs layer are analytically derived by applying double Fourier integral transforms to the general solutions and boundary conditions, which are consequently converted to the corresponding influence coefficients. The conjugate gradient method is used to obtain the unknown pressure distribution, while the discrete convolution–fast Fourier transform technique is applied to calculate the displacements and stresses of phonon and phason, electric potentials and electric displacements. Numerical results are given to reveal the influences of layer thickness, material parameters and loading conditions on the contact behavior. The obtained 3D contact solutions are not only helpful for further analysis and understanding of the coupling characteristics of phonon, phason and electric field, but also provide a reference basis for experimental analysis and material development.

A novel isogeometric analysis enriched element for a V-notched one-dimensional hexagonal piezoelectric quasicrystal bi-material

A novel isogeometric analysis enriched element for accurate evaluation of singularities arising at the interface in V-notched one-dimensional hexagonal piezoelectric quasicrystals (PQCs) under anti-plane loading is developed. By taking the advantages of IGA and symplectic methodology, an enriched element centered at the notch tip is constructed by analytical symplectic eigensolutions. Explicit expressions of notch intensity factors and singular multi-physical fields within enriched element are obtained simultaneously. A comparison study between numerical predictions and analytical solutions is performed and excellent agreements are observed. Furthermore, the notch intensity factors for different V-notched PQC bi-material are presented and discussed in detail.

Axial guided wave characteristics in functionally graded one-dimensional hexagonal piezoelectric quasi-crystal cylinders

In one-dimensional hexagonal piezoelectric quasi-crystals, there exist the phonon–phason, electro–phonon, and electro–phason couplings. Therefore, the phonon–phason coupling and piezoelectric effects on axial guided wave characteristics in one-dimensional hexagonal functionally graded piezoelectric quasi-crystal (FGPQC) cylinders are investigated by utilizing the Legendre polynomial series method. The dispersion curves and cut-off frequencies are illustrated. Wave characteristics in three hollow cylinders with different quasi-periodic directions are comparatively studied. Some new wave phenomena are revealed: the phonon–phason coupling and piezoelectric effects on the longitudinal and torsional phonon modes (N = 0) vary as the quasi-periodic direction changes; the phonon–phason coupling effect on flexural–torsional modes in the r-, z-FGPQC hollow cylinders, and on flexural–longitudinal modes in ϑ-FGPQC hollow cylinders increases as N increases. The corresponding results obtained in this work lay the theoretical foundation for the design and manufacture of piezoelectric transducers with high resolution and energy-conversion efficiency.

Surface effects on a mode-III reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals

To effectively reduce the field concentration around a hole or crack, an anti-plane shear problem of a nano-elliptical hole or a nano-crack pasting a reinforcement layer in a one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) is investigated subject to remotely mechanical and electrical loadings. The surface effect and dielectric characteristics inside the hole are considered for actuality. By utilizing the technique of conformal mapping and the complex variable method, the phonon stresses, phason stresses, and electric displacements in the matrix and reinforcement layer are exactly derived under both electrically permeable and impermeable boundary conditions. Three size-dependent field intensity factors near the nano-crack tip are further obtained when the nano-elliptical hole is reduced to the nano-crack. Numerical examples are illustrated to show the effects of material properties of the surface layer and reinforced layer, the aspect ratio of the hole, and the thickness of the reinforcing layer on the field concentration of the nano-elliptical hole and the field intensity factors near the nano-crack tip. The results indicate that the properties of the surface layer and reinforcement layer and the electrical boundary conditions have great effects on the field concentration of the nano-hole and nano-crack, which are useful for optimizing and designing the microdevices by PQC nanocomposites in engineering practice.

Anti-plane fracture analysis of 1D hexagonal piezoelectric quasicrystals with the effects of damage due to materials degradation

Most fracture mechanics analysis is based on the assumptions of prefabricated crack and ideal undamaged body. In practical, a cracked structure contains microstructural defects so that the material damage is considered to exist in the body rather than an ideal defect-free body. In this paper, it is the first time to take the material damage into account to the constitutive equation of quasicrystals, and the anti-plane fracture analysis of the one-dimensional hexagonal piezoelectric quasicrystals is carried out to obtain the more actual solutions. The continuum damage mechanics theory is used to define the damage variable and evolution law of the micro-voids’ growth in the quasicrystals. Semi-qualitative analysis of damage effects on the stress intensity factors are conducted, in which the stress intensity factors and J-integral are solved by using the boundary element method. In a case study, the fields of stress and displacement along the crack or distributed on the plate are obtained. The comparisons between the presented numerical results and the analytical results have good agreement, which verifies the accuracy of the presented solutions. The numerical results suggest that fracture failure analysis for quasicrystals engineering applications should consider the effective (actual) stress intensity factors calculated by the effective stress instead of nominal stress intensity factors.

Antiplane Fracture Problem of Three Nanocracks Emanating from an Electrically Permeable Hexagonal Nanohole in One-Dimensional Hexagonal Piezoelectric Quasicrystals

Based on the Gurtin-Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the electrically permeable boundary condition with surface effect is established, and the antiplane fracture problem of three nanocracks emanating from a hexagonal nanohole in one-dimensional hexagonal piezoelectric quasicrystals with surface effect is studied. The exact solutions of the stress intensity factor of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate are obtained under the two electrically permeable and the electrically impermeable boundary conditions. The numerical examples show the influence of surface effect on the stress intensity factors of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate under the two boundary conditions. It can be seen that the surface effect leads to the coupling of the phonon field, phason field, and electric field, and with the decrease of cavity size, the influence of surface effect is more obvious.

Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals**Project supported by the National Key R&D Program of China (Grant No. 2017YFC1405605), the Innovation Youth Fund of the Ocean Telemetry Technology Innovation Center of the Ministry of Natural Resources, China (Grant No. 21k20190088), the Natural Science Foundation of Inner Mongolia, China (Grant No. 2018MS01005), and

By constructing a new conformal mapping function, we study the surface effects on six edge nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional (1D) hexagonal piezoelectric quasicrystals under anti-plane shear. Based on the Gurtin–Murdoch surface/interface model and complex potential theory, the exact solutions of phonon field, phason field and electric field are obtained. The analytical solutions of the stress intensity factor of the phonon field, the stress intensity factor of the phason field, the electric displacement intensity factor and the energy release rate are given. The interaction effects of the nano-cracks and nano-hole on the stress intensity factor of the phonon field, the stress intensity factor of the phason field and the electric displacement intensity factor are discussed in numerical examples. It can be seen that the surface effect leads to the coupling of phonon field, phason field and electric field. With the decrease of cavity size, the influence of surface effect is more obvious.

Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part I: Theoretical formulations

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.

Analysis of 3D planar crack problems of one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part II: Numerical approach

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are carried out systematically. Part II of the work aims to develop a general numerical approach to study 3D planar crack problems in 1D hexagonal piezoelectric QC media. Based on the theoretical formulations presented in Part I, a direct and effective method is proposed to derive the Green's functions for point extended displacement discontinuities (EDDs). These Green's functions are presented explicitly by a series of potential functions in a compact form. Using the superposition principle, the Green's functions for uniformly distributed EDDs over the crack elements are obtained. Related element solutions are used to construct the numerical approach, known as EDD boundary element method, for 1D hexagonal piezoelectric QCs. The proposed numerical method can be applicative to many complicated planar crack problems, such as multiple cracks, and cracks with non-uniform loadings, for 3D media composed of 1D hexagonal piezoelectric QCs with thermal effect. A comparison of the results obtained from the theoretical solutions given in Part I of the work with those obtained from the numerical method proposed here validates of the present investigation.

Interaction of a Screw Dislocation with Interface and Wedge-Shaped Cracks in One-Dimensional Hexagonal Piezoelectric Quasicrystals Bimaterial

Interaction of a screw dislocation with wedge-shaped cracks in one-dimensional hexagonal piezoelectric quasicrystals bimaterial is considered. The general solutions of the elastic and electric fields are derived by complex variable method. Then the analytical expressions for the phonon stresses, phason stresses, and electric displacements are given. The stresses and electric displacement intensity factors of the cracks are also calculated, as well as the force on dislocation. The effects of the coupling constants, the geometrical parameters of cracks, and the dislocation location on stresses intensity factors and image force are shown graphically. The distribution characteristics with regard to the phonon stresses, phason stresses, and electric displacements are discussed in detail. The solutions of several special cases are obtained as the results of the present conclusion.