Love wave propagation in one-dimensional piezoelectric quasicrystal multilayered nanoplates with surface effects

The exact solutions for the propagation of Love waves in one-dimensional(1D)hexagonal piezoelectric quasicrystal(PQC)nanoplates with surface effects are derived.An electro-elastic model is developed to investigate the anti-plane strain problem of Love wave propagation.By introducing three shape functions,the wave equations and electric balance equations are decoupled into three uncorrelated problems.Satisfying the boundary conditions of the top surface on the covering layer,the interlayer interface,and the matrix,a dispersive equation with the influence of multi-physical field coupling is provided.A surface PQC model is developed to investigate the surface effects on the propagation behaviors of Love waves in quasicrystal(QC)multilayered structures with nanoscale thicknesses.A novel dispersion relation for the PQC structure is derived in an explicit closed form according to the non-classical mechanical and electric boundary conditions.Numerical examples are given to reveal the effects of the boundary conditions,stacking sequence,characteristic scale,and phason fluctuation characteristics on the dispersion curves of Love waves propagating in PQC nanoplates with surface effects.

Green's functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper,we obtain Green’s functions of two-dimensional(2D)piezoelectric quasicrystal(PQC)in half-space and bimaterials.Based on the elastic theory of QCs,the Stroh formalism is used to derive the general solutions of displacements and stresses.Then,we obtain the analytical solutions of half-space and bimaterial Green’s functions.Besides,the interfacial Green’s function for bimaterials is also obtained in the analytical form.Before numerical studies,a comparative study is carried out to validate the present solutions.Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force,the line dislocation,the line charge,and the phason line force.As a result,the coupling effect among the phonon field,the phason field,and the electric field is prominent,and the butterfly-shaped contours are characteristic in 2D PQCs.In addition,the changes of material parameters cause variations in physical quantities to a certain degree.

The interaction between a screw dislocation and a wedge-shaped crack in one-dimensional hexagonal piezoelectric quasicrystals

Based on the fundamental equations of piezoelasticity of quasicrystal material,we investigated the interaction between a screw dislocation and a wedge-shaped crack in the piezoelectricity of one-dimensional hexagonal quasicrystals.Explicit analytical solutions are obtained for stress and electric displacement intensity factors of the crack,as well as the force on dislocation.The derivation is based on the conformal mapping method and the perturbation technique.The influences of the wedge angle and dislocation location on the image force are also discussed.The results obtained in this paper can be fully reduced to some special cases already available or deriving new ones.

Anti-plane Analysis of a Circular Hole withThree Unequal Cracks in One-dimensionalHexagonal Piezoelectric Quasicrystals

This paper employes variable function method and the technique of conformal mappingto discuss the anti-plane problem of a circular hole with three unequal cracksin a one-dimensional （1D） hexagonal piezoelectric quasicrystal. Based on the piezoelectricityfundamental equations of quasicrystal materials and the symmetry of1D hexagonal quasicrystal and its linear piezoelectricity effect, 1D hexagonal quasicrystalcontrol equations of anti-plane problem are derived. Applying Cauchyintegral formula, the analytical expressions for the crack tip filed intensity factorsare presented with the assumption that the crack are electrical impermeable andelectrical permeable. With the variation of the hole-size and the crack length, someof the new model of crack are obtained. In the absence of the electric load, theresults match with the classical ones. The numerical results indicate the effects ofgeometric parameters on the field intensity factors. It is verified that the horizontalcrack length and the circle radius can easily promote crack growth. Researchon such issues will provide reliable theoretical value for the engineering materialspreparation and application.

Static deformation of a multilayered one-dimensional hexagonal quasicrystal plate with piezoelectric effect

Quasicrystals （QCs） are sensitive to the piezoelectric （PE） effect. This paper studies static deformation of a multilayered one-dimensional （1D） hexagonal QC plate with the PE effect. The exact closed-form solutions of the extended displacement and traction for a homogeneous piezoelectric quasicrystal （PQC）plate are derived from an eigensystem. The general solutions for multilayered PQC plates are then obtained using the propagator matrix method when mechanical and electrical loads are applied on the top surface of the plate. Numerical examples for several sandwich plates made up of PQC, PE, and QC materials are provided to show the effect of stacking sequence on phonon, phason, and electric fields under mechanical and electrical loads, which is useful in designing new composites for engineering structures.

Effective electroelastic constants for three-phase confocal elliptical cylinder model in piezoelectric quasicrystal composites

A three-phase confocal elliptical cylinder model is proposed to analyze micromechanics of one-dimensional hexagonal piezoelectric quasicrystal （PQC） compos- ites. Exact solutions of the phonon, phason, and electric fields are obtained by using the conformal mapping combined with the Laurent expansion technique when the model is subject to far-field anti-plane mechanical and in-plane electric loadings. The effective elec- troelastic constants of several different composites made up of PQC, quasicrystal （QC）, and piezoelectric （PE） materials are predicted by the generalized self-consistent method. Numerical examples are conducted to show the effects of the volume fraction and the cross-sectional shape of inclusion （or fiber） on the effective electroelastic constants of these composites. Compared with other micromechanical methods, the generalized self- consistent and Mori-Tanaka methods can predict the effective electroelastic constants of the composites consistently.

Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals

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.

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.

Study on a straight dislocation in an icosahedral quasicrystal with piezoelectric effects

An electro-elastic analysis is performed on an icosahedral quasicrystal with piezoelectric effects containing a straight dislocation. The closed-form expressions for the elastic and electric fields are obtained using the extended Stroh formalism. The effects of piezoelectric constant on the phonon displacement, phason displacement, and electric potential are discussed in detail.

Interaction between a screw dislocation and an elliptical hole with two asymmetrical cracks in a one-dimensional hexagonal quasicrystal with piezoelectric effect

The interaction between a screw dislocation and an elliptical hole with two asymmetrical cracks in a one-dimensional(1 D) hexagonal quasicrystal with piezoelectric effect is considered. A general formula of the generalized stress field, the field intensity factor, and the image force is derived, and the special cases are discussed. Several numerical examples are given to show the effects of the material properties and the dislocation position on the field intensity factors and the image forces.

Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials

The present paper is exposed theoretically to the influence on the dynamic stress intensity factor （DSIF） in the piezoelectric bi-materials model with two symmet- rically permeable interracial cracks near the edges of a circular cavity, subjected to the dynamic incident anti-plane shearing wave （SH-wave）. An available theoretical method to dynamic analysis in the related research field is provided. The formulations are based on Green＇s function method. The DSIFs at the inner and outer tips of the left crack are obtained by solving the boundary value problems with the conjunction and crack- simulation technique. The numerical results are obtained by the FORTRAN language program and plotted to show the influence of the variations of the physical parameters, the structural geometry, and the wave frequencies of incident wave on the dimensionless DSIFs. Comparisons with previous work and between the inner and outer tips are con- cluded.

Fundamental solutions of critical wedge angles for one-dimensional piezoelectric quasicrystal wedge

Two problems of a one-dimensional(1D)piezoelectric quasicrystal(QC)wedge are investigated,i.e.,the two sides of the wedge subject to uniform tractions and the wedge apex subject to the concentrated force.By virtue of the Stroh formalism and Barnett-Lothe matrices,the analytical expressions of the displacements and stresses are derived,and the generalized solutions for the critical wedge angles are discussed.Numerical examples are given to present the mechanical behaviors of the wedge in each field.The results indicate that the effects of the uniform tractions and the concentrated force on the phonon field displacement are larger than those on the phason field.

Singularities of Three-Dimensional Cubic Piezoelectric Quasicrystal Composite Wedges and Spaces

In this paper,the planar problems of three-dimensional(3D)cubic piezoelectric quasicrystal composite wedges and spaces are investigated.The study focuses on the singular behaviors of interface corner and interface crack of composite wedges and spaces.To research the stress singularities,the stress function is assumed to have the exponential form.Based on the Stroh formalism and Barnett–Lothe matrices,we derive a crucial matrix concerned with material properties and wedge angle and obtain the transcendental equation determining the singular orders by simple multiplication of the crucial matrix.Numerical examples of the singular orders are given for some general cases including single,bi-material,and tri-material wedges and spaces under different boundary conditions.The correctness of numerical results is verified by comparison with the existing results of piezoelectric material.Numerical results show that the phonon field,phason field,electric field,material properties,and boundary conditions have great influences on singularities.

一维六方压电准晶双材料椭圆孔边不对称裂纹断裂行为

基于一维六方压电准晶理论,利用复变函数方法研究了一维六方压电准晶双材料椭圆孔边不对称裂纹的反平面问题。利用保角变换将椭圆孔孔边裂纹问题转化为直裂纹问题,结合Stroh公式求解得到电不可通和电可通两种条件下场强度因子和能量释放率的解析表达式。数值算例结果显示:一维六方压电准晶双材料界面Griffith裂纹比椭圆孔边裂纹更易促进裂纹扩展;考虑相位子场可使能量释放率变化更显著,表明压电准晶材料相比压电材料更脆。此研究可为压电准晶元件的设计和制备提供理论依据。

含切口的压电准晶组合结构界面断裂分析的辛-等几何耦合方法

发展了一种适用于含有切口的压电准晶/压电晶体/弹性体三材料组合结构界面断裂问题的高精度的半数值半解析方法.首先,通过引入Hamilton体系建立了三材料组合结构的Hamilton对偶方程,将原问题在传统Lagrange体系下的高阶偏微分控制方程转化为低阶常微分方程组.其次,通过分离变量法求解问题对应的辛本征值和本征解,将各物理场变量利用辛级数展开形式表示.最后,将辛级数与等几何分析方法相结合,获得了辛-等几何耦合列式,直接求得切口尖端附近奇异物理场及其强度因子的解析表达式.

准晶及压电材料黏附接触力学研究进展

作为具备多场耦合特性的先进功能材料,准晶及压电材料目前已被广泛应用于各种微纳器件与结构.在微纳尺度下,系统的力学行为将由各种表面力(如静电力、毛细力以及范德华力等)引起的黏附效应主导.近年来,准晶及压电材料在微纳尺度下的黏附接触行为获得越来越多学者的关注和研究.本文在简要介绍经典黏附接触理论的基础上,综述了准晶及压电材料相关黏附接触力学问题最新研究进展,其中包括准晶材料的理论和实验研究以及压电材料理论研究.最后对未来准晶及压电材料黏附接触力学问题研究进行了展望.

一维六方压电准晶中正n边形孔边快速传播裂纹反平面问题的解析解

目的研究压电效应下一维六方准晶材料中含正n边形孔边快速传播裂纹的反平面问题。方法运用动力学基本理论和Schwarz-Christoffel变换技术,在电非渗透型和电渗透型2种边界条件下研究。结果给出了裂纹以速度v传播时Ⅲ裂纹的动态应力强度因子和电位移强度因子的解析解。结论当v→0时,所得结果退化为静力学问题,这对工程结构和材料的制备与应用有一定的指导意义。

Green’s functions for infinite planes and half-planes consisting of quasicrystal bi-materials

This paper deals with the combination of point phonon and phason forces applied in the interior of infinite planes and half-planes of 1D quasicrystal bi-materials. Based on the general solution of quasicrystals, a series of displacement functions are adopted to obtain Green's functions for infinite planes and bi-material planes composed of two half-planes in the closed form, when the two half-planes are supposed to be ideally bonded or to be in smooth contact. Since the physical quantities can be readily calculated without the need of performing any transform operations, Green's functions are very convenient to be used in the study of point defects and inhomogeneities in the quasicrystal materials.

一维六方压电准晶双材料界面共线裂纹问题

利用复变函数理论中的解析延拓、奇性主部分析和推广的Liouville定理,求解了一维六方压电准晶双材料在集中载荷作用下界面共线裂纹反平面弹性问题.导出了含有一条和两条有限长界面裂纹的封闭解,同时给出了裂纹尖端场强度因子(包含声子场和相位子场应力强度因子和电位移强度因子)的表达式.数值算例分析了外荷载与耦合系数之比对裂纹尖端场强度因子变化规律的影响.从数值结果中可以看出,当裂纹长度增加时,裂纹尖端场强度因子随之增加;应力强度因子随双材料耦合系数之比的增大而增大,电位移强度因子几乎不变;不同载荷作用下,裂纹尖端场强度因子随着裂纹长度改变时的变化趋势也不尽相同.研究结果可为压电准晶双材料的设计和制备提供一定的理论参考.

电半渗透边界条件下一维六方压电准晶纳米弯折裂纹反平面问题研究

利用复变函数方法及G-M(Gurtin-Murdoch)表/界面理论,研究了电半渗透条件下一维六方压电准晶纳米弯折裂纹反平面问题.引入弯折裂纹的保角映射,得到电半渗透条件下场强度因子以及能量释放率的解析解.数值分析了纳米弯折裂纹的介电常数、弯折角度、外载荷对无量纲场强度因子以及能量释放率的影响.结果表明:介电常数对相位子场无量纲应力强度因子以及电位移无量纲强度因子影响较为显著;场强度因子均随着裂纹与坐标轴夹角的增大而减小;随着外载荷的增加,场强度因子以及能量释放率有显著的变化且变化趋势具有差异性.本文可以为准晶及其复合材料的断裂行为的进一步研究奠定基础,也可以为纳米准晶材料的制备及应用提供一定的理论依据.