HYPERSINGULAR INTEGRAL EQUATIONS FOR PERIODIC ARRAYS OF PLANAR CRACKS IN A PERIODICALLY-LAYERED ANISOTROPIC ELASTIC SPACE UNDER ANTIPLANE SHEAR STRESS

1991MRSubjectClassification75M25,45E991IntroductionDuringthelasttenyearsorsojmanyresearchersinappliedmathematicsandmechanicshaveshownasurginginterestinformulatinglinearcrackproblemsillterlllsofsystel-alsofHadamardfillite-part(hypersingular)integralequations,e.g.Ioakimidis['],Lin'kovandMogilevskaya[']andAnal'].Anadvantageofsuchaformulationisthatthe11nkllowllfllnctionsaredirectlyrelatedtothejlllxlpillthedisplacementsacrossoppositeera(:kfaces.Oncetheyaredeterlttillied,crackparaliietersofinter…     

The Basic Mixed Problem for an Anisotropic Elastic Body

The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.     

Contact problems for two anisotropic half-planes with slits

The problem of a stressed state in a nonhomogeneous infinite plane consisting of two different anisotropic half-planes and having slits of finite number on the interface line is investigated. It is assumed that the difference between the displacement and stress vector values is given on the interface line segments; on the edges of the slits we have the following data: boundary values of stress vector (problem of stress) or displacement vector values on one side of the slits, and stress vector values on the other side (mixed problem). Solutions are constructed in quadratures.     

条状功能梯度材料中偏心裂纹对反平面简谐波的散射问题

利用Schmidt方法研究了条状功能梯度材料中偏心裂纹对反平面简谐波的散射问题,裂纹垂直于条状功能梯度材料的边界．通过Fourier变换,问题可以转换为对一对未知变量是裂纹表面位移差的对偶积分方程求解．为了求解对偶积分方程,把裂纹表面的位移差展开为Jacobi多项式级数形式,进而得到了功能梯度参数、裂纹位置以及入射波频率对应力强度因子影响的规律．     

对称迭层矩形板的平面应力分析

常用的对称迭层板为各向异性板．根据平面应力问题的基本方程精确地用应力函数解法求得了各向异性板的一般解析解．推导出平面内应力和位移的一般公式,其中积分常数由边界条件来决定．一般解包括三角函数和双曲函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式解,它能满足4个角的边界条件．因此一般解可用以求解任意边界条件下的平面应力问题．以4边承受均匀法向和切向载荷以及非均匀法向载荷的对称迭层方板为例,进行了计算和分析．     

Elastic Analysis of Anisotropic Functionally Graded Rotating Disks With Non-Uniform Thicknesses北大核心CSCD

研究了任意梯度变化的变厚度各向异性转动圆盘的弹性问题.假设圆盘绕刚性轴匀速转动,其材料性能和厚度沿径向任意梯度变化.考虑圆盘在中心转轴处受位移约束,外侧自由,根据各向异性转动圆盘的平衡微分方程,得到关于径向应力的Fredholm积分方程,继而通过对Fredholm积分方程进行数值求解,得到结构的位移场和应力场.对具体梯度变化情况仅需代入相应梯度变化进行求解即可.数值算例部分,通过假设厚度、弹性模量等参数为特殊的幂函数形式,将由Fredholm积分方程求出的数值解与对应的精确解进行对比,以及针对常见的Voigt模型,将由该方法算得的数值解和ANSYS有限元计算结果进行对比,验证了该方法的准确性和精度.其次,针对Voigt模型,重点分析了厚度变化、材料性能梯度参数、各向异性度等对应力场和位移场的影响.提出了针对材料性能和厚度沿径向呈任意梯度变化的圆盘结构弹性分析方法,将为优化功能梯度圆盘的结构和材料参数、有效调整构件应力分布、提高结构安全性,提供强有力的工具;算例分析结果对功能梯度圆盘在复杂条件下的结构安全设计有重要的理论指导意义.     

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.      

Asymptotic solutions of coupled dynamic problems of thermoelasticity for isotropic plates

The asymptotic method of solving boundary-value problems of the theory of elasticity for anisotropic strips and plates is used to solve coupled dynamic problems of thermoelasticity for plates, on the faces of which the values of the temperature function and the values of the components of the displacement vector or the conditions of the mixed problem of the theory of elasticity are specified. Recurrence formulae are derived for determining the components of the displacement vector, the stress tensor and for the temperature field variation function of the plate.     

压电压磁复合材料中界面裂纹对弹性波的散射

利用Schmidt方法分析了压电压磁复合材料中可导通界面裂纹对反平面简谐波的散射问题．经过富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程A·D2在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式．数值模拟分析了裂纹长度、波速和入射波频率对应力强度因子、电位移强度因子、磁通量强度因子的影响A·D2从结果中可以看出,压电压磁复合材料中可导通界面裂纹的反平面问题的应力奇异性形式与一般弹性材料中的反平面问题应力奇异性形式相同．     

Mixed boundary-value problems of thermoelasticity for anisotropic-in-plan inhomogeneous toroidal shells

Problems of thermoelasticity for an anisotropic-in-plan inhomogeneous thin toroidal shell are solved by asymptotic integration of the equations of the three-dimensional problem of the theory of an anisotropic inhomogeneous solid for various boundary conditions. Recurrence formulae are derived for the components of the asymmetric stress tensor and the displacement vector. An example is given.     

Wavelets on Regular Surfaces Generated by Stokes Potentials

By means of the limit and jump relations of potential theory with respect to the Stokes equations the framework of a tensorial wavelet approach on a regular (Lyapunov-) surface is established. The setup of a multiresolution analysis is defined by interpreting the kernel functions of the limit and jump integral operators as scaling functions on the regular surfaces. The distance of the parallel surface to the surface under consideration thereby represents the scale level in the scaling function. Tensorial scaling functions and wavelets show space localizing properties. Thus, they can be used to represent vector fields locally on a regular surface. Furthermore, these functions can be used as ansatz functions for the discretization of Fredholm integral equations of the second kind which result from the Stokes boundary-value problems with respect to a regular surface. By this, scaling functions and wavelets enable us to give a multiscale representation of the solution of the Stokes problem. This representation will be demonstrated in a concrete example. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

含有直线状裂纹正交异性板的平面问题的应力函数

本文的解析对象为含有一与主轴呈任意角度直线状裂纹的无限大正交异性板的平面问题.采用加权积分法导出了能够表现裂纹尖端附近有限应力集中特征的应力函数.这样的计算模型消除了裂纹尖端的奇异性,可以比较真实地反映非金属材料微裂区的力学行为.     

Solvability of the linearized problem on the motion of a drop in a liquid flow

In the paper one establishes the solvability in the anisotropic Sobolev-Slobodetskii spaces of the linear problem, generated by the problem of the nonstationary motion of a drop in a fluid medium. In the formulation of the problem one takes into account the surface tension, which occurs in the noncoercive integral term in the conditions for the jump of the normal stresses. In the general case the velocity vector need not be solenoidal but its divergence must be represented in a special form. The proof of the solvability is carried out first in the Sobolev-Slobodetskii spaces and is based on a priori estimates for the solutions in these spaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 171, pp. 53–65, 1989.     

Analytic solution to the problem of the temperature jump in a gas with rotational degrees of freedom

An exact solution is obtained for the first time for the problem of the temperature jump in a gas with allowance for internal (rotational) degrees of freedom. The treatment is based on a model collision integral proposed by the authors. The problem reduces to the solution of a boundary-value problem for a linear vector transport equation with matrix kernel. Separation of the variable leads to a characteristic equation for which eigenvectors are found in a space of generalized functions and the eigenvalue spectrum is investigated. An expansion of the solution to the problem with respect to eigenvectors of the continuous and discrete spectra is established. On the basis of the conditions of solvability of the vector Riemann-Hilbert boundary-value problem which arises in the process of the proof, an exact (in closed form) expression is obtained for the temperature jump.Moscow Pedagogical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 530–540, June, 1993.     

不同功能梯度压电压磁层状介质中共线界面裂纹的动态性能分挝

对不同功能梯度压电压磁层状介质中,共线界面裂纹对简谐应力波作用下的动态问题,进行了分析.经Fourier变换,使问题的求解转换为求解以裂纹面上位移间断为未知量的三重对偶积分方程,三重对偶方程可以采用Schmidt方法来求解,进而分析了功能梯度参数、入射波频率和层状介质厚度对应力、电位移和磁通量强度因子的影响.     

State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.     

Non-classical interface problems for piecewise homogeneous anisotropic elastic bodies

Two non-classical model interface problems for piecewise homogeneous anisotropic bodies are studied. In both problems on the contact surface jumps of the normal components of displacement and stress vectors are given. In addition, in the first problem (Problem H) the tangent components of the displacement vectors are given from both sides of the contact surface, while in the second one (Problem G) the tangent components of the stress vectors are prescribed on the same surface. The existence and uniqueness theorems are proved by means of the boundary integral equation method, and representations of solutions by single layer potentials are established. In the investigation the general approach of regularization of the first kind of integral equations is worked out for the case of two-dimensional closed smooth manifolds. An equivalent global regularizer operator is constructed explicitly in the form of a singular integro-differential operator.     

On the local solvability of evolution equations for the interface of two filtering fluids in the class of analytic functions

We consider a system of equations that describes the evolution of the interface of two fluids in the two-dimensional problem of their combined filtration in a porous homogeneous medium. This system contains a nonlinear integro-differential equation with a singular integral over an unknown contour together with a Fredholm integral equation of the second kind for the jump of the velocity vector potential on the movable boundary. We prove the unique solvability of such a system on a small time interval for the case in which the initial shape is defined parametrically and the functions describing the dependence of the coordinates of the point on the line on the parameter admit analytic continuations into the complex plane.     

The three-dimensional problem for an anisotropic plate of arbitrary thickness

By expanding the components of the displacement vector in a certain system of functions of the transverse coordinate, we reduce the solution of the three-dimensional problem of the theory of elasticity of an anisotropic body to a series of two-dimensional problems. To determine the displacements we obtain a system of differential equations of infinite order with two independent variables. We show how to pass from the infinite system to a series of finite systems depending on the form of the external forces. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 11–19.     

Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.