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.     

一维六方准晶中螺型位错偶极子与直线纳米裂纹的干涉效应

关于准晶中螺型位错偶极子与直线纳米裂纹干涉效应的问题，提出了一种可获得精确解的新方法。首先，将Gurtin-Murdoch表/界面理论推广运用到准晶材料中的纳米裂纹问题，建立了裂纹表面上含声子场、相位子场的应力边界模型；然后，利用复变函数中的共形映射方法，将直线裂纹问题化为单位圆孔问题；再借助于柯西积分方法获得了问题的封闭解；最后，数值结果表明：当裂纹的尺寸减小到纳米量级时，表面效应将对声子场和相位子场中的应力场、像力及像力偶矩产生显著的影响，从而改变其力学性能，即裂纹尺寸越小，表面效应越强，声子场和相位子场中的应力场、像力及像力偶矩的数值变化越大；螺型位错偶极子力偶臂的方向对像力的极值和自身平衡亦有较大影响。     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

十次对称二维准晶中的圆弧形裂纹

采用复变函数法研究了在远场承受均布声子场机械载荷作用下十次对称二维准晶中穿透周期轴的一个圆弧形裂纹.推导了表征声子场和相位子场的4个复应力函数的解析表达式,并由复应力函数获得了两个裂尖处声子场和相位子场应力强度因子、裂面张开位移以及能量释放率的表达式.     

Elastic field near the tip of an anticrack in a decagonal quasicrystalline material

We investigate the elastic field near the tip of an anticrack in a homogeneous decagonal quasicrystalline material subject to plane strain deformations. The phonon and phason stresses exhibit a square root singularity at the anticrack tip. Two realvalued phonon stress intensity factors and two real-valued phason stress intensity factors are introduced to scale four separate modes of deformation. We obtain four analytic functions which completely characterize the induced phonon and phason stresse...     

Bending analyses of 1D orthorhombic quasicrystal plates

The meshless Petrov–Galerkin method (MLPG) is applied to plate bending analysis in 1D orthorhombic quasicrystals (QCs) under static and transient dynamic loads. The Bak and elasto-hydrodynamic models are applied for phason governing equation in the elastodynamic case. The phason displacement for the orthorhombic QC in the first-order shear deformation plate theory depends only on the in-plane coordinates on the mean plate surface. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding this node. The coupled governing partial differential equations are satisfied in a weak-form on small fictitious subdomains. The spatial variations of the phonon and phason displacements are approximated by the moving least-squares (MLS) scheme. After performing the spatial MLS approximation, a system of ordinary differential equations (ODEs) for nodal unknowns is obtained. The system of the ODEs of the second order is solved by the Houbolt finite-difference scheme. Our numerical examples demonstrate clearly the effect of the coupling parameter on both static and dynamic phonon/phason deflections.     

A DECAGONAL QUASICRYSTAL WITH AN ARC-SHAPED CRACK

A decagonal quasicrystal,which is weakened by an arc-shaped crack penetratingthrough the solid in the period direction,and which is subjected to remote uniform phonon stresses,is investigated by applying the complex variable method which is just developed by the authors.It is found that the phonon and phason stresses near the crack tips exhibit inverse square rootsingularities.The four complex stress functions characterizing the phonon and phason fields arederived.Explicit expressions for the phonon and phason stress intensity factors,crack openingdisplacements and energy release rate are also presented.     

SH波对一维六方准晶中直裂纹的散射

分析了SH波对一维六方准晶中直裂纹的散射问题。利用积分变换技术,结合Copson方法,通过求解对偶积分方程,得到声子场和相位子场应力、位移及裂纹尖端动应力强度因子的解析表达式。通过数值算例讨论了裂纹长度、入射角和入射波频率对标准动应力强度因子的影响,此研究在工程材料应用中有一定的参考价值。     

含直对称裂纹的一维六方压电准晶对SH波的散射Scattering of SH wave by a linear symmetric crack at one-dimensional piezoelectric hexagonal quasicrystals

利用积分变换技术,结合Copson方法,研究了含直线型对称裂纹的一维六方压电准晶对SH波的散射问题。通过求解对偶积分方程,得到声子场、相位子场应力、位移及电场电位移分量的解析解。定义了裂纹尖端应力强度因子及电位移强度因子,给出了电非渗透性条件下应力强度因子及电位移强度因子的解析解。此研究结果对压电准晶材料的工程应用有一定的理论价值。     

Solutions for hydrodynamics of 5- and 10-fold symmetry quasicrystals

This study carries out a complete analysis of time-space solution of hydrodynamics of pentagonal/decagonal quasicrystals. The behaviors of wave propagation for phonons and diffusion for phasons and coupling between phonon-phason fields are explored explicitly. Comprehensive discussion on physical time-space variations of all hydrodynamic field variables of the alloy quasicrystals is given. The computational specimen is simple, convenient in testing computational results, and provides a possibility that is easy to test experimentally. The quantitative results of mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, viscosity stresses, and their time-space variations help us understand the motion of solid quasicrystals in a hydrodynamic condition (long-wavelength and low-frequency). The analysis presented in this paper can be used for octagonal and dodecagonal quasicrystals and is easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. Some problems explored by the computational results are also discussed.     

Elastic field in an infinite medium of one-dimensional hexagonal quasicrystal with a planar crack

This present work is concerned with planar cracks embedded in an infinite space of one-dimensional hexagonal quasicrystals. The potential theory method together with the general solutions is used to develop the framework of solving the crack problems in question. The mode I problems of three common planar cracks (a penny-shaped crack, an external circular crack and a half-infinite crack) are solved in a systematic manner. The phonon and phason elastic fundamental fields along with some important parameters in crack analysis are explicitly presented in terms of elementary functions. Several examples are given to show the applications of the present fundamental solutions. The validity of the present solutions is discussed both analytically and numerically. The derived analytical solutions of crack will not only play an important role in understanding the phonon–phason coupling behavior in quasicrystals, but also serve as benchmarks for future numerical studies and simplified analyses.     

T应力对立方准晶中含十字交叉型裂纹的影响

论文研究了立方准晶中含十字交叉型裂纹尖端附近的T应力.采用了积分变换法,将相关边值问题转换为求解奇异积分方程,得到声子场和相位子场的T应力,且可以表示为裂纹张开位移和施加在无穷远处的载荷之和.裂纹张开位移在确定声子场和相位子场的T应力中起着主导作用.同时强调了T应力在立方准晶脆性断裂中的作用.此外,还研究了裂纹臂长度比...     

A steady line heat source in a decagonal quasicrystalline half-space

The problem of a steady line heat source within a decagonal quasicrystalline half-space, in which the boundary is traction-free, is investigated in detail by applying the complex variable technique. The line heat source is infinitely long in the period direction, then the plane strain condition prevails. The four complex stress functions are derived. It is found that the determination of the complex stress functions is independent of the thermal boundary conditions. The stress fields induced by the heat source are explicitly given based on the complex stress functions. Furthermore, we consider, both qualitatively and quantitatively, the distribution of σ₁₁ on the boundary y = 0 of the half-space to demonstrate the influence of the phonon–phason coupling constant R on the stress field.     

An exact solution for a multilayered two-dimensional decagonal quasicrystal plate

By extending the pseudo-Stroh formalism to two-dimensional decagonal quasicrystals, an exact closed-form solution for a simply supported and multilayered two-dimensional decagonal quasicrystal plate is derived in this paper. Based on the different relations between the periodic direction and the coordinate system of the plate, three internal structure cases for the two-dimensional quasicrystal layer are considered. The propagator matrix method is also introduced in order to treat efficiently and accurately the multilayered cases. The obtained exact closed-form solution has a concise and elegant expression. Two homogeneous quasicrystal plates and a sandwich plate made of a two-dimensional quasicrystal and a crystal with two stacking sequences are investigated using the derived solution. Numerical results show that the differences of the periodic direction have strong influences on the stress and displacement components in the phonon and phason fields; different coupling constants between the phonon and phason fields will also cause differences in physical quantities; the stacking sequences of the multilayer plates can substantially influence all physical quantities. The exact closed-form solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and laminated plates. The numerical results can also be employed to verify the accuracy of the solution by numerical methods, such as the finite element and difference methods, when analyzing laminated composites made of quasicrystals.     

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.     

Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory

Based on the modified couple-stress theory, the three-dimensional(3D)bending deformation and vibration responses of simply-supported and multilayered twodimensional(2D) decagonal quasicrystal(QC) nanoplates are investigated. The surface loading is assumed to be applied on the top surface in the bending analysis, the tractionfree boundary conditions on both the top and bottom surfaces of the nanoplates are used in the free vibration analysis, and a harmonic concentrated point loading is applied o...     

Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip

The anti-plane fracture problem for a finite crack in a one-dimensional hexagonal quasicrystal strip is analyzed. By using Fourier transforms, the mixed boundary value problems are reduced to the dual integral equations. The solution of the dual integral equations is then expressed by the complete elliptic integrals of the first and the third kinds. The expressions for stress, strains, displacements and field intensity factors of the phonon and phason fields near the crack tip are obtained exactly. The path-independent integral derived by a conservation law equals the energy release rate, which can be used as a fracture criterion for a mode III fracture problem.     

Indentation on a one-dimensional hexagonal quasi-crystal half-space by an elliptic indenter

Meccanica - We investigate the contact between a one-dimensional (1D) hexagonal quasi-crystal (QC) half-space and a rigid elliptic indenter. The three-dimensional (3D) phonon–phason fields in...     

一维六方准晶层合简支梁自由振动与屈曲的精确解

伪Stroh型公式能够将多场耦合材料的控制方程转化为线性特征系统来求解,从而获得多层结构简支边界条件的精确解.本文利用伪Stroh型公式,研究一维六方准晶层合简支梁的自由振动和屈曲问题,通过传递矩阵法,获得准晶层合梁自由振动固有频率与临界屈曲载荷的精确解.通过与已有梁的剪切变形理论结果比较,验证了本文伪Stroh型公式的正确性和有效性.通过数值算例,分析由两种不同准晶材料组成的三明治层合梁的叠层方式、高跨比、层厚比及层数对梁的固有频率、临界屈曲载荷及其模态的影响规律.结果表明,叠层顺序和梁的高跨比、层厚比对准晶层合梁的自由振动固有频率和临界屈曲载荷有很大影响,可通过调整梁的几何尺寸和叠层顺序得到准晶层合梁的最佳固有频率和临界屈曲载荷.本文给出的精确解可为工程上研究准晶梁的各种数值解法和实验方法提供理论参考.