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。     

刚体的平面运动和旋轮线

讨论了作平面运动的薄板上一个质点在不同条件下的旋轮线轨迹,并讨论了这些不同形状的旋轮线是怎样由圆轮的纯滚动形成的.     

最速降线问题解充分性的证明

利用可动边界的变分公式,通过广义环路积分导出Hilbert不变积分,在此基础上说明了最速降线问题解的充分性.     

功能梯度旋转圆板的热弹性固有振动

在考虑热因素及旋转运动条件下,针对金属-陶瓷功能梯度圆板的固有振动问题进行研究.给出随温度变化且材料组分沿厚度方向按幂律分布的材料物性参数,依据热弹性理论得到圆板的能量关系式.基于哈密顿原理建立旋转金属-陶瓷功能梯度圆板热弹性动力学方程.采用伽辽金法得到边界约束下圆板的自由振动方程,确定了静挠度及固有振动频率.基于数值计算,得到系统固有频率值随体积分数指数、转速和温度等参量的变化曲线,讨论了静挠度变化规律及动力系统的奇点稳定性问题.结果表明,固有频率随体积分数指数、材料表面温度以及转速的增加而减小.     

压电材料反平面应变状态的任意形状夹杂问题

应用复函数的Ｆａｂｅｒ级数展开方法，分析了含任意形状夹杂的压电材料反平面应变问题，给出了问题的复势函数解。利用这个解，具体讨论了椭圆形夹杂及其极限（几何方面与物理方面）问题。并给出了三角形、正方形夹杂的近似结果。其特例结果与早期工作一致     

弹性力学轴对称问题的有限元线法

给出了解弹性力学空间轴对称问题的有限元线法的基本理论。该法包括了２－４条结线的等参数单元，沿结线方向的两点边值问题采用插值矩阵法解之。算例表明，本法具有良好的收敛性和较高的计算精度。     

弱界面异质球形夹杂本征应变问题的解析解

本文研究了具有线弹簧弱界面的异质球形夹杂的本征应变问题，所采用的线弹簧界面模型既能界面的切线方向滑动，又能考虑界面的法线方向张开，根据叠加原理、原问题的弹性场可分成三部分；二部分由真实均匀本征应变所引起，另一部分由等效的非均匀本征应变所引起，后一部分则由虚拟的Ｓｏｍｉｇｌｉａｎａ位错场所产生。本文求得了等效非均匀本征应变和虚拟位错场的Ｂｕｒｇｅｒ矢量的解析表达式，进而确定的问题的弹性场。     

线弹性孔洞问题的一种半解析解

在研究材料的损伤行为中，常常要考虑到孔洞效应，本文利用一组能满足内椭圆孔应力自由边界条件的罗朗级数，建立与外边界相适应的泛函，通过变分求泛函驻值或极值来求得相应问题，推广到双周期排列孔洞模型可得到损伤孔洞模型的等效杨氏模量和等效泊松比。     

基于位移型控制方程的圆板热后屈曲分析

本文推导了圆板位移型热后屈曲方程，在设定挠度试函数后，由微分方程精确求出径向位移，然后用Galerkin法消去另一方程的残差，试函数由Legendre多项式构成，结果表明：本文方法是有效的，有关结果可供设计圆板时参考。     

弹性圆板下横观各向同性弹性地基的轴对称问题

将与板相关的各力学量展开为Fourier-Bessel级数,利用解析法对弹性圆板下横观各向同性弹性地基的轴对称问题进行了分析。这些级数中的待定系数由板的边界条件、板的控制方程、地基-板的相容条件加以确定。数值计算结果表明:横观各向同性地基的沉降比各向同性地基的沉降要小,而地基反力则变化不大,说明按各向同性地基进行分析偏于保守。该分析方法将原问题简化为求解代数方程组的问题,且该方法可以推广到板的一般弯曲及层状地基与板的相互作用问题。     

The Analytical Solution of the Bending Problem for an Inhomogeneous Noncircular Cylindrical Body

A technique is developed for the analytical solution of the bending problem for cylindrical bar-like bodies on the basis of a nonclassical iterative model. Calculated results for differently fixed bars under uniformly distributed loading demonstrate the convergence of the iterative process.     

无限介质中单个纳米涂层夹杂反平面问题的两个模型

采用零厚度界面模型和界面层模型研究了无限介质中单个纳米涂层圆柱形夹杂的反平面问题,利用复变函数方法获得了两种模型夹杂、涂层和基体内应力场的封闭解析解．研究表明,当界面层模型中的界面相厚度趋于零时,界面层模型可以解析地退化为零厚度界面模型．数值算例分析了界面模量不同取值时应力场的分布和应力的尺度依赖性．本文结果丰富了对纳米夹杂力学行为的认识,并可为直接采用零厚度界面模型有困难的纳米夹杂问题的研究提供有价值的参考．     

On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method

Stress analysis of an elliptical inhomogeneity in an infinite isotropic elastic plane is a classical elasticity problem, which is usually solved by means of the complex variable formulation. In this work, we demonstrate that an alternative method of solution for such a problem, via the equivalent inclusion method, may be more convenient and straightforward without recourse to complex potentials or curvilinear coordinates. The explicit analytical solution can be derived through simple algebraic manipulation, although the longitudinal eigenstrain component should be handled with care in the case of plane strain. Since the exterior Eshelby tensor for an elliptical inclusion is available in closed-form, the present study provides a full field stress solution expressed in Cartesian coordinates. Furthermore, the in-plane stress components are represented in terms of Dundurs’ parameters. The solution methodology and the convenient formulae of the stress concentration may be of practical use to the engineers in developing benchmarks for design evaluation.     

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.     

Solution of a Plane Hydrofracture Problem with Stress Contrast

A plane hydrofracture problem for the Khristianovich–Geertsma–de Klerk model is extended and solved in the case where a confining stress closing a fracture is not constant in the direction of its propagation. A method is developed for solving the problem with an arbitrary stress contrast. It is stated that the transition through a contact with positive (negative) contrast occurs with fracture arresting (acceleration), whose intensity is controlled by a dimensionless parameter derived from theoretical considerations and numerical results.     

压电半平面梯形夹杂的梯度特征应变问题

针对边界处自由和绝缘以及固定和绝缘两种不同的条件,分别计算分析了均匀和梯度特征应变下梯形夹杂内部和外部诱发产生的弹性场和电场,并且讨论了梯形夹杂角点处的奇异性．最后,计算了平均梯度特征应变为零时梯形夹杂内部产生的平均弹性场和电场．所得结果揭示了基体不同边界条件对诱导场的影响．     

On a Displacement Based Solution to an Antiplane Crack Problem for Inhomogeneous Anisotropic Elastic Materials

This paper employs a displacement based method to examine an antiplane crack problem for an inhomogeneous elastic material in which the elastic moduli vary continuously with the spatial coordinates. Expressions for the crack tip stress intensity factors and the crack displacement are obtained in terms of Chebyshev polynomials. Numerical results are obtained for some particular inhomogeneous elastic materials.     

Analytical Solution of the Problem of Displacement of a Gas Dissolved in a Melt by Plane and Spherical Crystallization Fronts

Selfsimilar solutions of the problem of displacement of a gas dissolved in a melt by plane and spherical crystallization fronts are found for the case where the crystal growth rate is inversely related to the square root of time. A criterion of the absence of gas displacement due to segregation is found. The problem for a plane crystallization front moving with a constant velocity is analytically solved by means of the Laplace transform method.     

Stress Analysis of an Elliptic Inclusion with Imperfect Interface in Plane Elasticity

In this paper, a semi-analytic solution of the problem associated with an elliptic inclusion embedded within an infinite matrix is developed for plane strain deformations. The bonding at the inclusion-matrix interface is assumed to be homogeneously imperfect. The interface is modeled as a spring (interphase) layer with vanishing thickness. The behavior of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface.Complex variable techniques are used to obtain infinite series representations of the stresses which, when evaluated numerically, demonstrate how the peak stress along the inclusion-matrix interface and the average stress inside the inclusion vary with the aspect ratio of the inclusion and a representative parameter h (related to the two interface parameters describing the imperfect interface in two-dimensional elasticity) characterizing the imperfect interface. In addition, and perhaps most significantly, for different aspect ratios of the elliptic inclusion, we identify a specific value (h ^*) of the (representative) interface parameter h which corresponds to maximum peak stress along the inclusion-matrix interface. Similarly, for each aspect ratio, we identify a specific value of h (also referred to as h ^* in the paper) which corresponds to maximum peak strain energy density along the interface, as defined by Achenbach and Zhu (1990). In each case, we plot the relationship between the new parameter h ^*and the aspect ratio of the ellipse. This gives significant and valuable information regarding the failure of the interface using two established failure criteria.     

Analytic Solution of the Problem of Additive Formation of an Inhomogeneous Elastic Spherical Body in an Arbitrary Nonstationary Central Force Field

We study the processes of additive formation of spherically shaped rigid bodies due to the uniform accretion of additional matter to their surface in an arbitrary centrally symmetric force field. A special case of such a field can be the gravitational or electrostatic force field. We consider the elastic deformation of the formed body. The body is assumed to be isotropic with elasticmoduli arbitrarily varying along the radial coordinate.We assume that arbitrary initial circular stresses can arise in the additional material added to the body in the process of its formation. In the framework of linear mechanics of growing bodies, the mathematical model of the processes under study is constructed in the quasistatic approximation. The boundary value problems describing the development of stress–strain state of the object under study before the beginning of the process and during the entire process of its formation are posed. The closed analytic solutions of the posed problems are constructed by quadratures for some general types of material inhomogeneity. Important typical characteristics of the mechanical behavior of spherical bodies additively formed in the central force field are revealed. These characteristics substantially distinguish such bodies from the already completely composed bodies similar in dimensions and properties which are placed in the force field and are described by problems of mechanics of deformable solids in the classical statement disregarding the mechanical aspects of additive processes.