十次对称二维准晶弹性半平面问题的复变函数方法

研究了集中力作用下二维十次对称准晶半平面弹性问题的复变函数方法.首先将Stroh公式推广到二维准晶中,这里保留了Stroh公式的本质特征,在此基础上,采用推广的Stroh公式给出了应力和位移的通解,结合边界条件,获得了应力和位移的解析表达式,为实际应用奠定了理论基础.表明复变函数方法是解决十次对称二维准晶复杂弹性边值问题的有力工具.     

平面十次对称准晶中Ⅱ型Griffith裂纹的求解

应用应力函数法,求解了二维十次对称准晶中的Ⅱ型Griffith裂纹问题。特点是把二维准晶的弹性力学问题分解成一个平面应变问题与一个反平面问题的叠加,通过引入应力函数,把平面应变问题的十八个弹性力学基本方程简化成一个八阶偏微分方程,并且求出了其在Ⅱ型Griffith裂纹情况的混合边值问题的解,所有的应力分量和位移分量都用初等函数表示出来,并且由此得出了准晶中Ⅱ型Griffith裂纹问题的应力强度因子和能量释放率。     

平面十次对称准晶中Ⅱ型Briffith裂纹的求解

应用应力函数法,求解了二维十次对称准晶中的Ⅱ型Griffith裂纹问题。特别是把二维准晶的弹性力学问题分解成一个平面应变问题与一个反平面问题的叠加,通过引入应力函数,把平面应变问题的十八个弹性力学基本方程简化成一个八阶偏微分方程,并且求出了其在Ⅱ型Griffith裂纹情况的混合边值问题的解,所有的应力分量和位移分量都用初等函数表示出来,并且由此得出了准晶中Ⅱ型Griffith裂纹问题的应力强度因子和能量释放率。     

带任意裂纹的一维六方准晶弹性半平面第一基本问题

研究了周期平面内含任意裂纹的一维六方准晶的弹性半平面第一基本问题.首先借助保角变换将半平面第一基本问题转化为单位圆内带任意裂纹的第一基本问题;再利用复变函数方法将求有界域内的弹性平衡问题转化为奇异积分方程的求解,并证明方程是唯一可解的.该问题的求解为研究工程断裂问题提供了理论方法.     

点群6一维六方准晶椭圆孔口与半无限裂纹反平面弹性问题的解析解

导出了点群6-维六方准晶反平面弹性问题的控制方程.利用复变方法,给出了点群6-维六方准晶在周期平面内的反平面弹性问题的应力分量以及边界条件的复变表示,通过引入适当的保角变换,研究了点群6-维六方准晶中带有椭圆孔口与半无限裂纹的反平面弹性问题,得到了椭圆孔口问题应力场的解析解,给出了半无限裂纹问题在裂纹尖端处的应力强度因子的解析解.在极限情形下,椭圆孔口转化为Griffith裂纹,并得到该裂纹在裂尖处的应力强度因子的解析解.当点群6-维六方准晶体的对称性增加时,其椭圆孔口与半无限裂纹的反平面弹性问题的解退化为点群6mm-维六方准晶带有椭圆孔口与半无限裂纹的反平面弹性问题的解。     

一维正方准晶中半无限裂纹问题的解析解

运用广义复变函数方法,通过构造适当的广义保角映射,研究了含有沿准周期方向穿透的半无限裂纹的一维正方准晶的反平面弹性问题,给出了在部分裂纹面上受均匀面外剪切时应力场和裂纹尖端应力强度因子的解析解.将此方法进一步推广到半无限裂纹垂直于一维正方准晶的准周期方向穿透的情形中,得到了相应的平面弹性问题的解析解.当准晶体的对称性增加时,还可以得出一维四方准晶相应问题的解析解.     

立方准晶压电材料的半空间问题北大核心CSCD

考虑了立方准晶压电材料的半空间问题.给出了反平面机械载荷和面内电载荷作用下立方准晶压电材料弹性问题的控制方程,结合半无限区域表面边界条件,利用算子理论和复变函数方法获得了立方准晶压电材料半空间问题一般解的表达式.基于一般解得到了集中线力作用下,半空间问题的声子场和相位子场的位移、应力以及电位移的解析表达式.     

一维六方准晶的两类周期接触问题

利用复变函数方法讨论了一维六方准晶非周期平面的两类周期接触问题,即无摩擦周期接触以及半平面粘结周期接触问题.利用Hilbert核积分公式,得到了两类周期接触问题封闭形式的解.对于无摩擦周期接触问题,给出了3种常见压头(周期直水平基底、周期直倾斜基底、周期圆基底)作用下接触应力的显式表达式;对于半平面粘结周期接触问题,给出了实际工程中常见的边界上有尖劈形周期位移情况下应力的解析表达式.当忽略相位子场的贡献时,结果与正交各向异性材料周期接触问题的相应结果一致.     

边界过原点的任意半平面中的RH边值问题

给出边界过原点的任意半平面中RH边值问题的提法,借助于解析函数的对称扩张将此问题转化为无穷直线上的Riemann边值问题,讨论了该问题的求解并得到该问题的一般解及可解性定理.     

十次对称二维准晶中的楔形裂纹问题

研究了集中力作用下十次对称二维准晶中的楔形裂纹问题.采用推广的Stroh公式给出了应力和位移的一般解,在此基础上,讨论了声子场环向应力和相位子场环向应力的变化规律.作为特例,给出了十次对称二维准晶半无限裂纹问题应力和位移的解析表达式.     

Periodic set of the interface cracks with contact zones

A closed form solution to the plane problem of the theory of elasticity for an infinite isotropic bimaterial space (plane) with a periodic set of the interface cracks with frictionless contact zones near its tips is obtained. By means of the complex function presentation the problem is reduced to the combined Dirichlet–Riemann boundary value problem for a sectionally–holomorphic function and solved exactly. The equations for the determination of the contact zone length as well as the closed form expressions for the stress intensity factors are carried out. The variation of the mentioned values with respect to the distance between the cracks is illustrated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Static and extending interface cracks with contact zones in anisotropic as well as in piezoelectric bimaterials

An interface crack with an electrically permeable and mechanically frictionless contact zone in a piezoelectric bimaterial under the action of a remote mixed mode mechanical loading as well as thermal and electrical fields is considered in the first part of this paper. By use of the matrix‐vector representations of thermal, mechanical and electrical fields via sectionally‐holomorphic functions the problems of linear relationships are formulated and solved exactly both for an electrically permeable and an electrically impermeable interface crack. For these cases the transcendental equations and clear analytical formulas are derived for the determination of the contact zone lengths and the associated fracture mechanical parameters. A plane strain problem for a crack with a frictionless contact zone at the leading crack tip extending stationary along an interface of two semi‐infinite anisotropic spaces with a subsonic speed under the action of various loading is considered in the second part of this paper. By introducing of a moving coordinate system connected with the crack tip and by using the formal similarity of static and propagating crack problems the combined Dirichlet‐Riemann boundary value problem is formulated and solved exactly for this case as well and a transcendental equation is obtained for the determination of the real contact zone length. It is found that the increase of the crack speed leads to an increase of the real contact zone length and the correspondent stress intensity factors which increase significantly for a quasi‐Rayleigh wave speed.     

圆形界面刚性线夹杂的平面问题

研究了圆弧形界面刚性线夹杂的平面弹性问题．集中力作用于夹杂或基体中的任意点,并且无穷远处受均匀载荷作用．利用复变函数方法,得到了该问题的一般解答．当只含一条界面刚性线夹杂时,获得了分区复势函数和应力场的封闭形式解答,并给出刚性线端部奇异应力场的解析表达式．结果表明,在平面荷载下界面圆弧形刚性线夹杂尖端应力场和裂纹尖端相似具有奇异应力振荡性．对无穷远加载的情况,讨论了刚性线几何条件、加载条件和材料失配对端部场的影响．     

Solutions of half-space and half-plane contact problems based on surface elasticity

Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.     

Mixed interface problems for anisotropic elastic bodies

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where, on one part of the interface, rigid contact conditions are give (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

A periodic system of electrically permeable cracks at the interface between two piezoelectric materials

We constructed a closed solution for a bimaterial plane consisting of two dissimilar piezoelectric half-planes with a periodic system of electrically permeable cracks at the interface between these materials. The presence of zones of smooth contact of the crack lips near their tips was taken into account. By representing the characteristics of electromechanical fields via piecewise analytic functions, we reduced the problem to a Dirichlet–Riemann periodic problem, which was solved exactly. As a result of numerical analysis of the derived solution, we studied the dependence of the relative length of the contact zones and stress intensity factors on the ratio between the crack length and period for different combinations of piezoelectric materials.