A class of nonlinear boundary value problems for the second-order<Emphasis Type="Italic">E</Emphasis><Subscript>2</Subscript> class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E₂ class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods. Foundation items: the National Natural Science Foundation of China (19671056); Shanghai Municipal Natural Scientific Foundation (99ZA14030, 01ZA14023); Jiangxi Provincial Natural Scientific Foundation (981102, 0211014) Biographies: LI Ming-zhong (1935−); XU Ding-hua (1967−)     

A WEIGHTED NORM INEQUALITY FOR THETA(t)-TYPE OSCILLATORY SINGULAR INTEGRALS

The theta (t) -type oscillatory singular integral operators has been discussed.With the non- negative locally integrable weighted function , the weighted norm inequality of theta (t)-type oscillatory singular integral operators is proved, and the weighted function hasreplaced by action of Hardy-Littlewood maximal operators several times .     

Singular perturbation of second-order nonlinear system with boundary perturbation

Ｗｅｃｏｎｓｉｄｅｒｉｎｔｈｉｓｐａｐｅｒｔｈｅｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｓｅｃｏｎｄ＿ｏｒｄｅｒｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｉｎｖｏｌｖｉｎｇｉｎｔｅｒｇｒａｌｏｐｅｒａｔｏｒεｙ″=ｆ(ｔ,ｙ,Ｔｙ,ε)ｙ′ ｇ(ｔ,ｙ,Ｔｙ,ε),(1)ｗｉｔｈｂｏｕｎｄａｒｙｐｅｒｔｕｒｂａｔｉｏｎｙ(ｔ,ε)｜ｔ=φ(ε)=α(ε),ｙ(ｔ,ε)｜ｔ=1 ψ(ε)=β(ε),(2)ｗｈｅｒｅε>0ｉｓａｓｍａｌｌｐａｒａｍｅｔｅｒ,ａｎｄφ(ε),ψ(ε)ａｒｅｂｏｔｈ,ｗｉｔｈｒｅｓｐｅｃｔｔｏε,ｓｕｆｆｉｃｉｅｎｔｌｙｓｍｏ…     

On the nonlinear Riemann problems for general first elliptic systems in the plane

IntroductionManyproblemsinmechanicsandmechanicalengineeringmaybeformulatedintoboundaryvalueproblemsforfirst_orsecond_orderellipticsystems.Lotsofscholarsandtheauthorsstudiedthem ( [1～ 6] ) .InthispaperwediscussthenonlinearRiemannproblemforgeneralsystemsofthefirst_orderlinearandquasi_linearequationsintheplane.1　TheNonlinearRiemannProblemforGeneralLinearEquationLetG beamultiplyconnecteddomaininthecomplexplaneEthatisboundedbyafinitenumberofclosed ,nonintersectingC1,αcurvesΓk,k=0 ,… ,mwit…     

Integral transform solution of multi-material structure with finite crack perpendicular to the interface

The plane elasticity problem for layered elastic systems containing a finite crack perpendicular to the interface is considered. To derive the singular integral equations. Fourier transform in conjunction with dislocation is used. The singular integral equation is solved with the Lobatto-Chebyshev method commonly applied to such problems. In order to have an idea about the usefulness of the method described, a two-layer structure which contains a cut parallel toh is considered.     

The scattering of general SH plane wave by interface crack between two dissimilar viscoelastic bodies

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored. This work was supported by the National Natural Science Foundation of China (No.19772064) and by the project of CAS KJ 951-1-20     

Helmholtz边界积分方程中奇异积分间接求解方法

提出了间接求解传统Helmholtz边界积分方程CBIE的强奇异积分和自由项系数,以及Burton-Miller边界积分方程BMBIE中的超强奇异积分的特解法。对于声场的内域问题,给出了满足Helmholtz控制方程的特解,间接求出了CBIE中的强奇异积分和自由项系数。对于声场外域对应的BMBIE中的超强奇异积分,按Guiggiani方法计算其柯西主值积分需要进行泰勒级数展开的高阶近似,公式繁复,实施困难。本文给出了满足Helmholtz控制方程和Sommerfeld散射条件的特解,提出了间接求出超强奇异积分的方法。推导了轴对称结构外场问题的强奇异积分中的柯西主值积分表达式,并通过轴对称问题算例证明了本文方法的高效性。数值结果表明,对于内域问题,采用本文特解法的计算结果优于直接求解强奇异积分和自由项系数的结果,且本文的特解法可避免针对具体几何信息计算自由项系数,因而具有更好的适用性。对于外域问题,两者精度相当,但本文的特解法可避免对核函数进行高阶泰勒级数展开,更易于数值实施。     

Weighted estimates for strongly singular integral operators with rough kernels

The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral operator is bounded from the Sobolev space to the Lebesgue space.     

半平面多边缘裂纹反平面问题的奇异积分方程

利用复变函数和奇异积分方程方法,求解弹性范围内半平面多边缘裂纹的反平面问题．提出了满足半平面边界自由的由分布位错密度表示的单边缘裂纹的基本解,此基本解由主要部分和辅助部分组成．将半平面多边缘裂纹问题看作是许多单边缘裂纹问题的叠加,建立了一组Cauchy型奇异积分方程．然后,利用半开型积分法则求解该奇异积分方程,得到了裂纹端处的应力强度因子．最后,给出了几个数值算例．     

Evaluation of strongly singular domain integrals for internal stresses in functionally graded materials analyses using RIBEM

An accurate evaluation of strongly singular domain integral appearing in the stress representation formula is a crucial problem in the stress analysis of functionally graded materials using boundary element method.To solve this problem,a singularity separation technique is presented in the paper to split the singular integral into regular and singular parts by subtracting and adding a singular term.The singular domain integral is transformed into a boundary integral using the radial integration method.Analytical expressions of the radial integrals are obtained for two commonly used shear moduli varying with spatial coordinates.The regular domain integral,after expressing the displacements in terms of the radial basis functions,is also transformed to the boundary using the radial integration method.Finally,a boundary element method without internal cells is established for computing the stresses at internal nodes of the functionally graded materials with varying shear modulus.     

FIXED POINTS OF INCREASING OPERATORS IN ORDERED SPACE WITH APPLICATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｖｅｓｔｉｇａｔｉｎｇｔｈｅｆｏｌｌｏｗｉｎｇｂｏｕｎｄａｒｙｐｒｏｂｌｅｍｉｎｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ :¨ｘ+ｆ(ｔ,ｘ(ｔ) ) =0 ,ａ<ｔ<ｂ,αｘ(ａ) -β ｘ(ａ) =0 ,γｘ(ｂ) +δ ｘ(ｂ) =0 ,( 1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,Δ=(ｂ-ａ)αγ+αδ+ βγ>0 .ｆ(ｔ,ｓ)ｍａｙｂｅｓｉｎｇｕｌａｒｉｎｔ =ａ ,ｂ.Ｉｔｈａｓｂｅｅｎｄｉｒｅｃｔｌｙｖｅｒｉｆｉｅｄｔｈａｔｘ(ｔ)ｉｓｔｈｅｓｏｌｕｔｉｏｎｏｆ( 1 )ｉｎＣ2 [ａ ,ｂ]ｉｆａｎｄｏｎｌｙｉｆｘ(…     

各向异性平面含斜裂纹的奇异积分方程方法

本文应用平面弹性复变方法，将无限各向异性平面中的任意斜裂纹问题归结为求解一组解析函数边值问题，通过构造适当的积分变换将边值问题转化为奇异积分方程，进而应用Lobotto-Chebyshev数值求积公式，求出该奇异积分方程的数值解，并得到了应力强度因子的近似表达式，最后，给出了一些实例的数值结果，对特例的数值结果与精确结果进行比较，吻合的很好。     

Numerical methods for solving singular integral equations obtained by fracture mechanical analysis of cracked wedge简

The numerical solutions to the singular integral equations obtained by the fracture mechanical analyses of a cracked wedge under three different conditions are considered. The three considered conditions are: (i) a radial crack on a wedge with a non-finite radius under the traction-traction boundary condition, (ii) a radial crack on a wedge with a finite radius under the traction-traction boundary condition, and (iii) a radial crack on a finite radius wedge under the traction-displacement boundary condition. According to the boundary conditions, the extracted singular integral equations have different forms. Numerical methods are used to solve the obtained coupled singular integral equations, where the Gauss-Legendre and the Gauss-Chebyshev polynomials are used to approximate the responses of the singular integral equations. The results are presented in figures and compared with those obtained by the analytical response. The results show that the obtained Gauss-Chebyshev polynomial response is closer to the analytical response.     

纤维增强复合材料圆柱型界面裂纹分析

以裂纹面上的位错函数为未知量将圆柱型界面裂纹问题化成一组奇异积分方程的求解问题．应用Ｍｕｓｋｈｅｌｉｓｈｖｉｌｉ的奇异积分方程理论，分析了圆柱型界面裂纹尖端应力场．针对裂纹尖端分别存在和不存在接触区两种情况，确定了裂纹尖端应力场的奇异性．利用数值方法计算了圆柱型界面裂纹尖端接触区尺寸对剪应力强度因子的影响．     

Singular integral equations and boundary element method of cracks in thermally stressed planar solids

Ｉｔｉｓｋｎｏｗｎｔｈａｔｍｏｓｔａｇｒｉｃｕｌｔｕｒａｌｐｒｏｄｕｃｔｓａｎｄｆｏｏｄｓａｒｅｐｒｏｃｅｓｓｅｄａｎｄｔｒａｎｓｐｏｒｔｅｄｕｎｄｅｒｃｅｒｔａｉｎｔｅｍｐｅｒａｔｕｒｅｃｏｎｄｉｔｉｏｎｓ,ａｎｄｔｈｅｓｔｒｕｃｔｕｒａｌｃｏｍｐｏｎｅｎｔｓａｌｓｏｗｏｒｋｕｎｄｅｒａｔｈｅｒｍａｌｅｎｖｉｒｏｎｍｅｎｔ.Ｔｅｍｐｅｒａｔｕｒｅｉｎｄｕｃｅｄｓｔｒｅｓｓｅｓｕｓｕａｌｌｙｌｅａｄｔｏｄａｍａｇｅｏｆｆｌａｗｅｄｓｏｌｉｄｓ.Ｔｈｕｓ,ｔｈｅｉｎｖｅｓｔｉｇａｔｉｏｎｏｆｔｈｅｃｒ…     

层状介质垂直界面有限裂纹问题的积分变换解

层状弹性材料包含垂直于界面有限裂纹时，可运用富里叶变换及引用位错密度函数，导出了反映裂纹尖端奇异性的奇异积分方程组，并使用Ｌｏｂａｔｔｏ－ｃｈｅｂｙｓｈｅｖ方法解此方程组，最后得到裂纹尖端应力强度因子，为检验方法的正确性，对某两层含裂实际结构进行了计算，结果是满意的。     

黏弹性材料界面裂纹应力场奇性分析

研究两半无限大黏弹性体间Griffith界面裂纹在简谐载荷作用下裂纹尖端动应力场的奇异特性.通过引入裂纹张开位移和裂纹位错密度函数,相应的混合边值问题归结为一组耦合的奇异积分方程.渐近分析表明裂尖动应力场的奇异特征完全包含在奇异积分方程的基本解中.通过对基本解的深入分析发现黏弹性材料界面裂纹裂尖动应力场具有与材料参数和外载荷频率相关的振荡奇异特性.以标准线性固体黏弹材料为例讨论了材料参数和载荷频率对奇性指数和振荡指数的影响.     

多个纵向环形界面裂纹对P波散射的动应力强度因子

研究多个纵向环形界面裂纹的Ｐ波散射问题。以裂纹面的位错密度函数为未知量,利用Ｆｏｕｒｉｅｒ积分变换,将问题归结为第二类奇异积分方程,然后通过数值求解,获得裂纹尖端的动应力强度因子。最后给出了双裂纹动应力强度因子随入射波频率变化的关系曲线。     

正交各向异性位势问题边界元法中几乎奇异积分的解析算法

几乎奇异积分的计算困难阻碍了边界元法的工程应用。本文针对二维正交各向异性位势问题边界元法中近边界点的几乎奇异积分,采用分部积分法,导出一种直接的解析计算公式。该解析公式可以精确计算线性单元上的几乎奇异积分。对二次单元,可将其细分为几个线性元,采用该解析公式近似计算其边界积分。当内点离当前积分单元较远时,仍保持常规高斯数值积分模式;而当内点离其较近时,因常规高斯积分结果失效,则采用该解析积分取代高斯数值积分。数值算例证明了该算法的有效性和精确性。二次元计算结果比线性元计算结果更精确。     

On the Solution of Mixed Problems in Linear Anti-plane Piezoelectricity

In the present paper we consider interior and exterior mixed boundary value problems of anti-plane shear in the static theory of linear piezoelectricity. Using the boundary integral equation method we reduce the problems to systems of singular integral equations with discontinuous coefficients to which the classical Nöether’s theorems on existence of the solution can be applied. This allows us to establish well-posedness results and to obtain integral solutions of the corresponding mixed boundary value problems for a rather general class of piezoelectric materials. Mathematics Subject Classifications (2000) 45E05, 45F15, 74F15.