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,...     

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...     

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。     

双周期圆柱形压电夹杂反平面问题的精确解及其应用

研究无限压电介质中双周期圆柱形压电夹杂的反平面问题.借鉴Eshelby等效夹杂原理,通过引入双周期非均匀本征应变和本征电场,构造了一个与原问题等价的均匀介质双周期本征应变和本征电场问题.利用双准周期Riemann边值问题理论,获得了夹杂内外严格的电弹性解.作为压电纤维复合材料的一个重要模型,预测了压电纤维复合材料的有效电弹性模量.     

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。     

横观各向同性介质中椭圆夹杂受非弹性剪切变形引起的弹性场的确定

本文求解了横观各向同性介质中椭圆夹杂内受非弹性剪切变形引起的弹性场。采用各向异性弹性力学平面问题的复变函数解法,结合保角变换,获得夹杂内应变能和基体内边界的应力分布和相应的应变能的表达式。进一步,根据最小应变能原理,获得表征夹杂平衡边界的两个特征剪切应变,从而得到了弹性场的解析解。通过应力转换关系,验证了应力解满足夹杂边界上法向正应力和剪应力的连续条件,表明了该解的正确性。本文解可用于复合材料断裂强度的分析中。     

纳米夹杂复合材料的有效反平面剪切模量研究

基于Gurtin-Murdoch表面/界面理论模型,利用复变函数方法,获得了考虑夹杂界面应力时夹杂/基体/等效介质模型的全场精确解,发展了能够预测纳米夹杂复合材料有效反平面剪切模量的广义自洽方法,给出了复合材料有效反平面剪切模量的封闭形式解。数值结果显示：当夹杂尺寸在纳米量级时,复合材料的有效反平面剪切模量具有尺度相关性,随着夹杂尺寸的增大,本文结果趋近于经典弹性理论的预测值;夹杂尺寸对于有效反平面剪切模量(本文结果)的影响范围要小于其对有效体积模量与剪切模量(各向同性材料)的影响范围;有效反平面剪切模量受夹杂的界面性能和夹杂刚度影响显著。     

半平面内多个纳米圆形夹杂的表/界面效应

基于Gurtin-Murdoch表/界面理论,采用边界积分法,讨论了各向同性的弹性半平面中含有任意多个纳米圆形夹杂问题,得到了受表面/界面影响的纳米复合结构的应力和位移的数值解.最后,给出了半平面中含有单个纳米孔洞和纳米夹杂的数值算例,分析了纳米界面存在对整个半平面结构应力场的影响.     

各向异性体内多个夹杂对反平面波的散射

本文导出了各向异性介质反平面剪切运动的基本解。在此基础上引用等效体力及二维亥维赛函数建立了内含多个任意形状夹杂(空洞)时散射位移场的积分方程。针对两个异质物情形运用Born近似理论讨论了散射远场。同时定义和推导了微分横截线,并给出了计算实例。     

实验—数值综合法确定受载复合材料板的变形场和损伤场

本文根据广义弹脆性损伤理论模型，用综合实验分析和数值计算的方法，确定受载复合材料板的形变场和损伤场。首先用云纹干涉法确定各受损单元的节点位移，再用有限元分析得到这些单元的真实应变和有效应变。由此计算受载各单元的损伤变量和有效弹性系数。最后，根据弹脆性材料的损伤本构关系确定受载复合材料板的真实应力场。     

New Green’s function for stress field and a note of its application in quantum-wire structures

The elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two-dimensional hydrostatic inclusion subjected to plane strain conditions. The stresses in such a hydrostatic inclusion can be effectively calculated by employing the Green’s functions developed by Downes and Faux, which tend to be more efficient than the conventional method based on the Green’s function for the displacement field. In this study, Downes and Faux’s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains: an explicit Green’s function solution, which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly, is obtained in a closed-form. Here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived Green’s functions. In the case of uniform eigenstrain, the formulae may be simplified to contour integrals by Green’s theorem. However, special care is required when Green’s theorem is applied for the interior field. The proposed Green’s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non-uniform eigenstrain. Two examples concerning circular inclusions are investigated. A linearly distributed eigenstrain is attempted in the first example, resulting in a linear interior stress field. The second example solves a circular thermal inclusion, where both the interior and exterior stress fields are obtained simultaneously.     

The determination of the elastic field of a polygonal star shaped inclusion

Recently we found that the elastic field is uniform in a pentagonal star (five-pointed star inclusion) [1], and in a triangular inclusion [2], when an eigenstrain is distributed uniformly in these inclusions. This result is similar to the famous result of Eshelby (1957) that the elastic field is uniform in an ellipsoidal inclusion in an infinitely body when an eigenstrain is distributed uniformly in the ellipsoidal inclusion. We also found that for a Jewish star (Star of David or six points star) or a rectangular inclusion subjected to a uniform eigenstrain, the stress field is not uniform in these inclusions. These results also hold for two dimensional plane strain cases. Furthermore these analytical results are confirmed experimentally by photoelasticity method. In this paper, we investigate a more general inclusion of an m-pointed polygonal inclusion subjected to the uniform eigenstrain. We conclude that the stress field is uniform when m is odd number. This conclusion agrees with the speculation made by B. Boley after the author's talk at Shizuoka [2].     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

周期分布多边形夹杂奇异性应力干涉问题的研究

采用一种新型的杂交元模型和一种单胞模型来解决周期分布多边形夹杂角部的奇异性应力相互干涉的问题。新型杂交元模型是基于广义Hellinger-Reissner变分原理建立的,其中奇异性应力场分量和位移场分量是采用有限元特征分析法的数值特征解得到的。使用当前的新型杂交元模型,只需要在夹杂角部邻域的周界上划分一维单元,避免了像传统有限元模型那样需要划分高密度二维单元。文中给出了代表奇异性应力场强度的夹杂角部广义应力强度因子数值解,并考虑材料属性、夹杂尺寸和夹杂位置关系的影响。算例中,考虑了夹杂和基体完全接合的情况,并给出了考核例。结果表明:当前模型能得到高精度数值解,且收敛性好;与传统有限元法和积分方程方法相比,该模型更具有通用性,为非均质材料的细观力学分析打下了基础。     

Circular inclusions in anti-plane strain couple stress elasticity

The solution for a circular inclusion with a prescribed anti-plane eigenstrain is derived. It is shown that the components of the Eshelby tensor within the inclusion, corresponding to a uniform eigenstrain, can be either uniform or non-uniform, depending on the imposed interface conditions. The stress amplification factors due to circular void or rigid inclusion in an infinite medium under remote anti-plane shear stress are calculated. The failure of the couple stress elasticity to reproduce the classical elasticity solution in the limit of vanishingly small characteristic length is indicated for a particular type of boundary conditions. The solution for a circular inhomogeneity in an infinitely extended matrix subjected to remote shear stress is then derived. The effects of the imposed interface conditions, the shear stress and couple stress discontinuities, and the relationship between the inhomogeneity and its equivalent eigenstrain inclusion problem are discussed.     

Uniformity of Stresses Within a Three-Phase Elliptic Inclusion in Anti-Plane Shear

In this paper, we show that a three-phase elliptic inclusion under uniform remote stress and eigenstrain in anti-plane shear admits an internal uniform stress field provided that the interfaces are two confocal ellipses. The exact closed-form solution is used to quantify the effect of the interphase layer on the residual stresses within the inclusion and the dependency of this effect on the aspect ratio of the elliptic inclusion. This revised version was published online in August 2006 with corrections to the Cover Date.     

The principle of equivalent eigenstrain for inhomogeneous inclusion problems

In this paper, based on the principle of virtual work, we formulate the equivalent eigenstrain approach for inhomogeneous inclusions. It allows calculating the elastic deformation of an arbitrarily connected and shaped inhomogeneous inclusion, by replacing it with an equivalent homogeneous inclusion problem, whose eigenstrain distribution is determined by an integral equation. The equivalent homogeneous inclusion problem has an explicit solution in terms of a definite integral. The approach allows solving the problems about inclusions of arbitrary shape, multiple inclusion problems, and lends itself to residual stress analysis in non-uniform, heterogeneous media. The fundamental formulation introduced here will find application in the mechanics of composites, inclusions, phase transformation analysis, plasticity, fracture mechanics, etc.     

Inclusion of arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric full plane

In this paper, an exact closed-form solution for the Eshelby problem of a polygonal inclusion with a graded eigenstrain in an anisotropic piezoelectric full plane is presented. For this electromechanical coupling problem, by virtue of Green’s function solutions, the induced elastic and piezoelectric fields are first expressed in terms of line integrals on the boundary of the inclusion. Using the line-source Green’s function, the line integral is then carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. Finally, the solution is applied to the semiconductor quantum wire (QWR) of square, triangle, circle and ellipse shapes within the GaAs (0 0 1) substrate. It is demonstrated that there exists significant difference between the induced field by the uniform eigenstrain and that by the linear eigenstrain. Since the misfit eigenstrain in most QWR structures is actually non-uniform, the present solution should be particularly appealing to nanoscale QWR structure analysis where strain and electric fields are coupled and are affected by the non-uniform misfit strain.     

Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

具有光滑界面的椭球夹杂的轴对称弹性场

本文在旋转椭球坐标系下,利用Papkovich—Neuber位移通解求解了具有光滑界面椭球夹杂由于均匀的特征应变引起的轴对称弹性场,与理想界面不同,在夹杂与基体界面不能经受剪应力而可自由滑动的情况下,解答只能是无穷级数形式,因此文中给出了数值算例。