双周期分布圆环形截面夹杂反平面问题的解析方法

研究含双周期分布圆环形截面弹性夹杂的无限大介质在远场均匀反平面应力下的弹性响应。通过在双周期圆环形区域内引入非均匀本征应变,将双周期非均匀介质问题转化为带有双周期非均匀本征应变的均匀介质问题,结合双周期函数和双准周期Riemann边值问题理论,获得了该问题弹性场的级数形式解答。作为一个应用,利用该解答预测了含双周期圆环形截面夹杂复合材料的有效纵向剪切模量。数值结果表明,在相同夹杂体积分数下,含圆环形截面夹杂的复合材料比含圆形截面夹杂的复合材料拥有更高的有效纵向剪切模量。     

任意形状热夹杂位移场的三角形单元离散算法

平面夹杂模型在纤维增强型复合材料中有广泛应用.复合材料内部通常含有不规则形状夹杂,而夹杂物的存在能严重影响材料的机械力学性能,往往导致应力集中及裂纹萌生等失效先兆.先前关于多边形夹杂的研究大多数关注受均匀本征应变下的应力/应变解,而对位移的分析较少. 基于格林函数方法和围道积分,本文给出了平面热夹杂边界线单元的封闭解析解,可方便应用于受任意分布本征应变的任意形状平面热夹杂位移场的数值计算.当夹杂受均匀本征应变时, 只需将该夹杂边界进行一维离散,因而本文方法可直接得出受均匀分布热本征应变的任意多边形夹杂位移场的封闭解析解.当夹杂区域存在非均匀分布本征应变时,可将该区域划分为足够小的三角形单元进行数值计算. 众所周知,应力应变场在多边形夹杂顶点处具有奇异性,容易导致数值计算上的处理困难及相应的数值稳定性问题; 然而本文工作表明,在多边形顶点处位移场是连续有界的, 因而数值稳定性较好.本文算法可以便捷高效地通过计算机编程实现. 文中给出的验证算例,均体现了本文离散方法的高精度、以及计算编程的鲁棒性.      

涂层对复合材料电-磁-弹性性能的影响

研究带界面层的双周期排列电磁弹性纤维复合材料在远场反平面机械载荷和面内电磁载荷作用下的电-磁-弹性响应,结合本征广义应变(本征应变、本征电场、本征磁场)的概念和双准周期Rimann边值问题理论为其发展了一个严格的解析分析方法.解答可用于对胞元内电磁弹性场量进行精细分析,也可对复合材料有效电磁弹性模量做出准确预测.以纯弹性环氧树脂涂层为例,分析了涂层对应力集中系数和有效电磁弹性模量的影响.本文结果可作为一个"基准",用以评价有限元等数值与近似方法的精度.     

任意形状热夹杂位移场的三角形单元离散算法

平面夹杂模型在纤维增强型复合材料中有广泛应用.复合材料内部通常含有不规则形状夹杂,而夹杂物的存在能严重影响材料的机械力学性能,往往导致应力集中及裂纹萌生等失效先兆.先前关于多边形夹杂的研究大多数关注受均匀本征应变下的应力/应变解,而对位移的分析较少.基于格林函数方法和围道积分,本文给出了平面热夹杂边界线单元的封闭解析解...     

双周期圆柱形夹杂纵向剪切问题的精确解

研究无限介质中矩形排列双周期圆柱形夹杂的纵向剪切问题．利用Eshelby等效夹杂理论并结合双周期与双准周期解析函数工具，为这类考虑夹杂相互影响的问题提供了一个严格又实用的分析方法，求得了问题的全场级数解．作为退化情形得到单夹杂问题的经典解答，双周期孔洞、双周期刚性夹杂及单行(列)周期弹性夹杂等问题也可作为特殊情况被解决．数值结果揭示了这类非均匀材料力学性质随微结构参数变化的规律．     

含有功能梯度界面相压电纤维复合材料的平面问题

基于复变函数理论和线弹性压电材料本构方程,研究了含有功能梯度界面相压电纤维复合材料平面问题的电-弹场.首先基于复变函数理论,对功能梯度界面相分层均匀化处理,然后将基体、分层均匀化后的界面相以及压电纤维的复势设定为含有待定系数的级数解形式,再通过边界条件建立相应的方程组求解复势中的待定系数,最后求得在各种载荷下压电复合材料的电-弹场.研究结果表明功能梯度界面相的材料属性对压电复合材料的电-弹场有重要的影响.     

平面任意形状等剪切模量异质夹杂问题的解析解

本文研究任意形状夹杂域在受到远端均匀荷载和均匀本征应变作用下的弹性场问题,其中基体和夹杂的材料不同但具有相同的剪切模量。利用等效理论将远端均匀荷载引起的扰动转化为等效均匀本征应变的作用,再利用K-M势函数表达扰动场问题的界面连续条件;借助于黎曼映射定理,用洛朗多项式将平面光滑闭合曲线外部区域映射到单位圆外部区域,借助柯西积分公式和Faber多项式求解了等剪切本征应变下夹杂和基体的K-M势函数的显式解析解,其中考虑了夹杂相对于基体的刚体位移。将得到的结果与相关文献的结果进行对比,表明了本论文的方法和结果是有效的和正确的。     

压电复合材料中的Eshelby夹杂问题

通过采用解析延拓和共形映射技术，获得了压电复合材料中有关Eshelby夹杂几个典型问题的精确弹性解答，即横观各向同性压电介质中任意形状的Eshelby夹杂与圆柱异相夹杂间相互作用；一般各向异性压电介质中任意形状的Eshelby夹杂与双压电材料所形成界面的相互作用．成功求解这些问题的关健在于构造一个辅助函数．与Ru所采用的方法不同，所引入的辅助函数在无穷远点不存在极点，从而使得所展开的分析更加自然合理．分析结果清楚地揭示出Eshelby夹杂的存在对压电复合材料机电耦合响应将产生不容被忽视的影响．很典型的一个例于是当一个Eshelby椭圆夹杂与圆柱异相夹杂相互作用时，每个夹杂体内部的应力场和电场都将是不均匀的；另一个例于是位于界面附近的Eshelby夹杂有可能是界面发生损伤的一个重要原因．     

等参三角形热夹杂的构造及位移场数值计算

在材料制备和机械设计中,局部温升是造成材料失效和故障形成的重要因素之一.依照微观力学中,采用热夹杂模型可以定量深入地揭示与局部温升所关联的力学机理.在过往的研究中,受均匀热本征应变的夹杂模型广受关注;而相关非均匀分布的热本征应变问题,因其理论推导复杂而研究不多.论文首先给出在平面无限域中,受线性分布热本征应变作用的多边形夹杂的位移场解析解.基于格林函数法和围道积分,推导边界线单元的位移响应封闭解,该解通过叠加可直接给出线性热本征应变作用下的任意多边形夹杂的解析表达式.受到有限元分析中等参单元思想的启发,论文进一步将这种“等参元”方法扩展至求解Eshelby夹杂问题中.在该研究中,三角形单元的本征应变插值公式与位置坐标变换式均使用了相同的形函数与节点参数,因而所构建的单元模型称为等参三角形夹杂模型.论文方法可便捷地用于处理受任何分布热本征应变的任意形状二维Eshelby夹杂问题.相较于传统的有限元分析,论文所构建的数值求解方案实施方便且优势明显：只需在夹杂域上进行三角形网格剖分、而无需在无限的基体域上划分网格,因而可以极大地提高前处理便捷性及计算效率.此外,论文所给出的多边形夹杂解析解,...     

基于界面层模型的双周期纳米夹杂复合材料反平面问题研究

利用复变函数方法并结合双准周期Riemann边值问题理论，获得了含双周期分布非均匀相（夹杂/界面层）的复合材料在远场均匀反平面应力下弹性场的全场解答．该解答可用于对纳米夹杂复合材料的应力进行分析，结合平均场理论也用于预测纳米夹杂复合材料的有效性能．计算结果表明：当夹杂尺度在纳米量级时，应力和有效反平面剪切模量具有明显的尺度依赖性，并且随着夹杂尺寸的增加，趋近于不考虑界面效应时的结果；界面层厚度和性能对应力和有效反平面剪切模量明显变化时所对应的夹杂尺度范围和趋近于无界面效应结果的快慢有显著影响；当界面厚度足够薄时，界面层模型可用于模拟零厚度界面情况．     

Heterogeneous linearly piezoelectric patches bonded on a linearly elastic body

In [1], we studied the response of a thin homogeneous piezoelectric patch perfectly bonded to an elastic body. Here we extend this study to the case of a very thin heterogeneous patch made of a periodic distribution of piezoelectric inclusions embedded in a linearly elastic matrix and perfectly bonded to an elastic body. Through a rigorous mathematical analysis, we show that various asymptotic models arise, depending on the electromechanical loading together with the relative behavior between the thickness of the patch and the size of the piezoelectric inclusions.     

Piezoelectric composites with periodic multi-coated inhomogeneities

A new, robust homogenization scheme for determination of the effective properties of a periodic piezoelectric composite with general multi-coated inhomogeneities is developed. In this scheme the coating does not have to be thin, the shape and orientation of the inclusion and coatings do not have to be identical, their centers do not have to coincide, their properties do not have to remain uniform, and the microstructure can be with the 2D elliptic or the 3D ellipsoidal inclusions. The development starts from the local electromechanical equivalent inclusion principle through the introduction of the position-dependent equivalent eigenstrain and electric field. Then with a Fourier series expansion and a superposition procedure, the volume-averaged equivalent eigenstrain and electric field for each phase are obtained. The results in turn are used in an energy equivalent criterion to determine the effective properties of the composite. In this model the interphase interactions in each multi-coated particle and the long-range interactions between the periodically distributed particles are fully accounted for. To demonstrate its wide range of applicability, we applied it to examine the properties of several periodic composites: (i) piezoelectric PZT spherical particles in a polymer matrix, (ii) continuous glassy fibers with thin PZT coating in an epoxy matrix, (iii) spherical PZT particles coated by thick or functionally graded piezoelectric layer, (iv) spheroidal voids coated with a thick non-piezoelectric layer in a PZT matrix, and (v) spherical piezoelectric inhomogeneities with eccentric, non-uniform thickness coating. The calculated results reflect the complex nature of interplay between the properties of core, matrix, and coating, as well as whether the coating is uniform, functionally graded, or eccentric. The accuracy of this new scheme is checked against the double-inclusion and other micromechanics models, and good agreement is observed.     

Transient response of piezoelectric composites caused by the sudden formation of localized defects

A continuum model is presented which is capable of generating the transient electroelastic field in piezoelectric composites of periodic microstructure, caused by the sudden appearance of localized defects. These defects are simulated by associating to every one of the ten piezoelectric parameters of the constituents a distinct damage variable. This procedure enables the modeling of localized cracks, soft and stiff inclusions and cavities. As a result, the constitutive equations of the piezoelectric phases appear in a specific form that includes eigen-electromechanical field variables which represent these defects. The method of solution is based on the combination of two distinct approach. In the first one, the representative cell method is employed according to which the periodic composite, which is discretized into several cells, is reduced to a problem of a single cell in the discrete Fourier transform domain. The resulting coupled elastodynamic and electric equations, initial, boundary and interfacial conditions in the transform domain are solved by employing a wave propagation in piezoelectric composite analysis which forms the second approach. The method of solution is verified by comparison with an analytical solution for the transient response of a piezoelectric material with a semi-infinite mode III-crack. Several applications are presented for the sudden formation of cracks in homogeneous and layered piezoelectric materials which are subjected to various types of electromechanical loading, and for the sudden appearance of a cavity. The effect of electromechanical coupling on the dynamic response is discussed.     

General solution for Eshelby’s problem of 2D arbitrarily shaped piezoelectric inclusions

Eshelby’s problem of piezoelectric inclusions arises sometimes in exploiting the electromechanical coupling effect in piezoelectric media. For example, it intervenes in the nanostructure design of strained semiconductor devices involving strain-induced quantum dot (QD) and quantum wire (QWR) growth. Using the extended Stroh formalism, the present work gives a general analytical solution for Eshelby’s problem of two-dimensional arbitrarily shaped piezoelectric inclusions. The key step toward obtaining this general solution is the derivation of a simple and compact boundary integral expression for the eigenfunctions in the extended Stroh formalism applied to Eshelby’s problem. The simplicity and compactness of the boundary integral expression derived make it much less difficult to analytically tackle Eshelby’s piezoelectric problem for a large variety of non-elliptical inclusions. In the present work, explicit analytical solutions are obtained and detailed for all polygonal inclusions and for the inclusions characterized by Jordan’s curves and Laurent’s polynomials. By considering the piezoelectric material GaAs (110), the analytical solutions provided are illustrated numerically to verify the coincidence between different expressions, and to clarify the jump across the boundary of the inclusion and the singularity around the corner of the inclusion.     

Asymptotic solutions by the successive disordering method

We develop the periodic componentmethod [1] and represent the solution of a stochastic boundary value elasticity problem for a random quasiperiodic structure with a given disordering degree of inclusions in the matrix via the deviations from the corresponding solution for a random structure with a smaller disordering degree. An example in which the tensor of elastic properties of a composite is calculated is used to illustrate the asymptotic and differential approaches of the successive disordering method. The asymptotic approach permits representing the quasiperiodic structure with a given chaos coefficient and the desired tensor of effective elastic properties as a result of small successive disordering of an originally ideally periodic structure of a composite with known tensor of elastic properties. In the differential approach, a differential equation is obtained for the tensor of effective elastic properties as a function of the chaos coefficient. Its solution coincides with the solution provided by the asymptotic approach. The solution is generalized to the case of piezoactive composites, and a numerical analysis of the effective properties is performed for a PVF (polyvinylidene fluoride) piezoelectric with various quasiperiodic structures on the basis of the cubic structure with spherical inclusions of a high-module elastic material.     

Piezocomposite with reciprocal polarization of oriented ellipsoidal inclusions and the matrix

We consider statistically homogeneous two-phase random piezoactive structures with deterministic properties of inclusions and the matrix and with random mutual location of inclusions. We present the solution of a coupled stochastic boundary value problem of electroelasticity for the representative domain of a matrix piezocomposite with a random structure in the generalized singular approximation of the method of periodic components; the singular approximation is based on taking into account only the singular component of the second derivative of the Green function for the comparison media. We obtain an analytic solution for the tensor of effective properties of the piezocomposite in terms of the solution for the tensors of effective properties of a composite with an ideal periodic structure or with the “statistical mixture” structure and with the periodicity coefficient calculated for a given random structure with its specific characteristics taken into account. The effective properties of composites with auxiliary structures (periodic and “statistical mixture”) are also determined in the generalized singular approximation by varying the properties of the comparisonmedium. We perform numerical computations and analyze the effective properties of a quasiperiodic piezocomposite with reciprocal polarization of oriented ellipsoidal inclusions and the matrix, the layered structures with reciprocal polarization of the layers [1] of a polymer piezoelectric PVF, and find their unique properties such as a significant increase in the Young modulus along the normal to the layers and in dielectric permittivities, the appearance of negative values of the Poisson ratio under extension along the normal, and an increase in the absolute values of the basic piezomoduli.     

Interacting circular inclusions in antiplanepiezoelectricity

The problem of multiple piezoelectric circular inclusions, which are perfectly bondedto a piezoelectric matrix, is analyzed in the framework of linear piezoelectricity. Both the matrixand the inclusions are assumed to possess the symmetry of a hexagonal crystal in the 6 mm classand subject to electromechanical loadings (singularities) which produce in-plane electric fieldsand out-of-plane displacement. Based upon the complex variable theory and the method ofsuccessive approximations, the solution of electric field and displacement field either in theinclusions or in the matrix is expressed in terms of explicit series form. Stress and electric fieldconcentrations are studied in detail which are dependent on the mismatch in the materialconstants, the distance between two circular inclusions, and the magnitude of electromechanicalloadings. It is shown that, when the two inclusions approach each other, the oscillatory behaviorof the stress and electric field can be induced in the inclusion as the matrix and the inclusions arepoled in the opposite directions. This important phenomenon can be utilized to build a verysensitive sensor in a piezoelectric composite material system. The present derived solution canalso be applied to the inclusion problem with straight boundaries. The problem associated withthree-material media under electromechanical sources is also considered.     

Dynamic fiber inclusions with elliptical and arbitrary cross-sections and related retarded potentials in a quasi-plane piezoelectric medium

A piezoelectric medium of transversely isotropic symmetry with continuous fiber inclusion parallel to the axis of symmetry is considered. The problem is equivalent to a two-dimensional ‘quasi-plane’ piezoelectric medium containing a 2D inclusion. The inclusion is assumed to undergo a spatially uniform δ(t)-type time domain transformation. The continuous fiber has elliptical, circular and arbitrary cross-sections. The solutions of the inclusion problem is expressed by scalar potentials. In the time domain two of these functions correspond to the retarded potential integrals of the inclusion. Their frequency domain representation which we shall call the ‘dynamic potentials of the inclusion’ are also considered. Integral formulae are derived for continuous fiber inclusions with elliptical cross-sections. Known closed-form solutions are reproduced for circular fibers. For fibers with arbitrary cross-sections a numerical method based on Gauss quadrature is applied. High accuracy and efficiency of the numerical method is confirmed. Characteristic superposition and runtime effects for the inclusions are found.     

The effective thermoelectroelastic properties of microinhomogeneous materials

The paper is concerned with composite materials which consist of a homogeneous matrix phase with a set of inclusions uniformly distributed in the matrix. The components of these materials are considered to be ideally elastic and exhibit piezoelectric properties. One of the variants of the self-consistent scheme, the Effective Field Method (EFM) is applied to calculate effective dielectric, piezoelectric and thermoelastic properties of such materials, taking into account the coupled electroelastic effects. At first the coupled thermoelectroelastic problem for a homogeneous medium with an isolated inclusion is solved. For an ellipsoidal inclusion and constant external field the solution of this problem is found in a closed analytic form. This solution is then used in the EFM to derive the effective thermoelectroelastic operator for the composite containing a random array of ellipsoidal inclusions. Explicit formulae for the electrothermoelastic constants are given for composites, reinforced by spheroidal inclusions.