奇点附近的各向异性塑性应力场

在奇点附近的理想塑性应力分量都只是θ的函数的条件下,利用平衡方程和Hill各向异性屈服条件,本文导出了反平面应变和平面应变两者奇点附近的各向异性塑性应力场的一般解析表达式。将这些一般解析表达式用于具体裂纹及有奇点的平面应变体,我们就得到Ⅰ型、Ⅱ型、Ⅲ型和Ⅰ-Ⅱ复合型裂纹尖端的各向异性塑性应力场以及有奇点的各向异性塑性平面应变体的极限载荷。     

Justification de la théorie non linéaire de Kirchhoff–Love,comme application d'une nouvelle méthode d'inversion singulièreJustification of the nonlinear Kirchhoff–Love theory,as the application of a new singular inverse method

In the framework of nonlinear elasticity, we consider a three-dimensional plate made of a St Venant–Kirchhoff isotropic and homogeneous material of thickness 2ε and periodic in the two other directions. By a change of scales, the problem can be mapped on a fixed open set, and seen as a nonlinear singular perturbation problem. We introduce a new singular inverse method. Applying this method, we prove that for fixed and small enough exterior forces, the three-dimensional displacement converges to the solution of the nonlinear Kirchhoff–Love theory of plate as the thickness 2ε tends to zero. The limit plate model contains in particular that of von Kármán. We also give a quantitative estimate of the convergence. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 615–620.     

Inplane Stress Singularities at the Corner of an Anisotropic Material Discontinuity

In the mechanics of composite laminates the local mechanical inplane fields at corners of anisotropic material discontinuities are of particular interest since they can have singular behavior. In the present study, the stress and strain fields in the local near field of such corners are investigated by an asymptotic analysis. The order of the singularity of these mechanical inplane fields are determined in closed‐form manner by use of the complex potential method based on Lekhnitskii's approach. Various different geometrical setups and material combinations of corners with material discontinuities are investigated with regard to their effect on the singular behavior of the mechanical fields present. These examples show that the order of singularity considered is clearly weaker than the typical crack tip singularity in fracture mechanics. Nevertheless, it may render the corner a critical location for the onset of failure. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic singular integrals in product spaces

In this paper, the authors introduce a class of product anisotropic singular integral operators, whose kernels are adapted to the action of a pair A := (A1, A2) of expansive dilations on R n and R m , respectively. This class is a generalization of product singular integrals with convolution kernels introduced in the isotropic setting by Fefferman and Stein. The authors establish the boundedness of these operators in weighted Lebesgue and Hardy spaces with weights in product A∞ Muckenhoupt weights on R n × R m . These results are new even in the unweighted setting for product anisotropic Hardy spaces.     

Weighted Lp Estimates for Maximal Commutators of Multilinear Singular Integrals

This paper is concerned with the pointwise estimates for the sharp function of two kinds of maximal commutators of multilinear singular integral operators T∑b^＊ and TПb^＊ which are generalized by a weighted BMO function b and a multilinear singular integral operator T, respectively. As applications, some commutator theorems are established.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

A new boundary element technique for solving plane problems of linear elasticity: improved theory and an application to fracture mechanics

An improved boundary element method for solving plane problems of linear elasticity theory is described. The method is based on the Muskhelishvili complex variable representation for the displacement and stress fields. The paper shows how to take account of symmetry about the x and/or y axes.The potential accuracy of the method is illustrated by its application to the calculation of stress intensity factors associated with cracks in both a square and a circular plate. The crack problem is solved using a Gauss-Chebyshev representation of a singular integral equation by a set of linear algebraic equations. The integral equation involves an analytic function which takes account of the presence of the external boundary. This function is determined directly using the boundary element method.Numerical results are believed to be more accurate than the existing published values which are quoted to four significant figures.     

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.     

A recursive methodology for the solution of semi-analytical rectangular anisotropic thin plates in linear bending

A new recursive methodology is introduced to solve anisotropic thin plates bending problems, which is based on three concepts: (a) the plate differential operator additively decomposed obeying a material constitutive hierarchy; (b) the plate displacement field expanded into an infinite series and (c) an homotopy-like scheme applied to determine each term of the series. The pb-2 Rayleigh–Ritz Method is adopted to construct the solution space. Convergence conditions are presented and related to the differential operator decomposition and material’s anisotropy degree. Different cases of geometry, loading and boundary conditions were studied using the methodology and excellent agreement with available solutions was found.     

The Littlewood-Offord problem and invertibility of random matrices

We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n−1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum k_∑akXk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p.     

Edge detection from truncated Fourier data using spectral mollifiers

Edge detection from a finite number of Fourier coefficients is challenging as it requires extracting local information from global data. The problem is exacerbated when the input data is noisy since accurate high frequency information is critical for detecting edges. The noise furthermore increases oscillations in the Fourier reconstruction of piecewise smooth functions, especially near the discontinuities. The edge detection method in Gelb and Tadmor (Appl Comput Harmon Anal 7:101–135, 1999, SIAM J Numer Anal 38(4):1389–1408, 2000) introduced the idea of “concentration kernels” as a way of converging to the singular support of a piecewise smooth function. The kernels used there, however, and subsequent modifications to reduce the impact of noise, were generally oscillatory, and as a result oscillations were always prevalent in the neighborhoods of the jump discontinuities. This paper revisits concentration kernels, but insists on uniform convergence to the “sharp peaks” of the function, that is, the edge detection method converges to zero away from the jumps without introducing new oscillations near them. We show that this is achievable via an admissible class of spectral mollifiers. Our method furthermore suppresses the oscillations caused by added noise.     

On the L 2-extension of twisted holomorphic sections of singular hermitian line bundles

We prove a sharp Ohsawa–Takegoshi–Manivel type L ²-extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds.     

Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials

According to the linear theory of elasticity, there exists a combination of different orders of stress singularity at a V-notch tip of bonded dissimilar materials. The singularity reflects a strong stress concentration near the sharp V-notches. In this paper, a new way is proposed in order to determine the orders of singularity for two-dimensional V-notch problems. Firstly, on the basis of an asymptotic stress field in terms of radial coordinates at the V-notch tip, the governing equations of the elastic theory are transformed into an eigenvalue problem of ordinary differential equations (ODEs) with respect to the circumferential coordinate θ around the notch tip. Then the interpolating matrix method established by the first author is further developed to solve the general eigenvalue problem. Hence, the singularity orders of the V-notch problem are determined through solving the corresponding ODEs by means of the interpolating matrix method. Meanwhile, the associated eigenvectors of the displacement and stress fields near the V-notches are also obtained. These functions are essential in calculating the amplitude of the stress field described as generalized stress intensity factors of the V-notches. The present method is also available to deal with the plane V-notch problems in bonded orthotropic multi-material. Finally, numerical examples are presented to illustrate the accuracy and the effectiveness of the method.     

Optimal embedding and sharp estimates of the continuity envelope for generalized Bessel potentials

In the theory of function spaces it is an important problem to describe the differential properties for the convolution u = G * f in terms of the behavior of kernel near the origin, and at the infinity. In our paper the differential properties of convolution are characterized by their modulus of continuity of order k ∈ N in the uniform norm. The kernels of convolution generalize the classical kernels determining the Bessel and Riesz potential. They admit non-power behavior near the origin. The order-sharp estimates are obtained for moduli of continuity of the convolution in the uniform norm as well as for continuity envelope function of generalized Bessel potentials. Such estimates admit sharp embedding theorems into a Calderon space and imply estimates for the approximation numbers of the embedding operator.     

具有抛物线边界的二维弹性介质的Green函数

文章求解了具有抛物线边界的二维弹性介质的两种Ｇｒｅｅｎ函数，一种是自由边界问题，另一种是刚性边界问题。我们还求得了当抛物线边界退化成半无限裂纹或半无限刚性裂纹时裂纹尖端的奇异场，得到了集中力作用于边界的基本解，这个基本解使得我们可以通过沿边界积分确定任意分布荷载的弹性解．     

On Singular Trudinger–Moser Type Inequalities for Unbounded Domains and Their Best Exponents

Generalizations of the Trudinger–Moser inequality to Sobolev spaces with singular weights are considered for any smooth domain Ω???? ^N . Furthermore, we show that the resulting inequalities are sharp obtaining the best exponents.     

Analysis of functionally graded plates using higher order shear deformation theory

This work addresses a static analysis of functionally graded material (FGM) plates using higher order shear deformation theory. In the theory the transverse shear stresses are represented as quadratic through the thickness and hence it requires no shear correction factor. The material property gradient is assumed to vary in the thickness direction. Mori and Tanaka theory (1973) [1] is used to represent the material property of FGM plate at any point. The thermal gradient across the plate thickness is represented accurately by utilizing the thermal properties of the constituent materials. Results have been obtained by employing a C° continuous isoparametric Lagrangian finite element with seven degrees of freedom for each node. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side-thickness, side–side) on deflection and temperature are investigated. Effect of skew angle on deflection and axial stress of the plate is also studied. Effects of material constant n on deflection and the temperature distribution are also discussed in detail.     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

Stress Singularities in Viscoelastic Media 2. Plane-Strain Stress Singularities at Corners

The stress singularities that evolve at the corner of a notchedviscoelastic angular plate subject to mode I deformation isdiscussed when prescribed, but arbitrary, displacements aresymmetrically applied to both radial edges of the sector. Thesolution procedure, based on Laplace and Mellin transforms (withtransform parameters p and s, respectively), leads to an eigenvalueproblem in the complex p-plane, which is dependent on Poisson'sratio and characterizes the singular behaviour of the stressfields. Although simple solutions to the transcendental eigenequationare not available, the real-time evolution of the stress concentrationsis obtained by monitoring a particular branch of the eigenvalueequation as it moves in the complex p-plane. A correspondingpath is traced in an appropriately cut strip in the s-plane,in which solutions to the eigenequation are single-valued. Analyticcontinuation in the p-plane thus allows the Laplace and Mellininversions to be performed and the real-time behaviour of theplane-stress components to be expressed as contour integralswithin the strips in the complex s-plane. Cast in this form,the stress components are evaluated numerically when the viscoelasticmaterial is represented as a standard linear solid. Their dependenceon the angular variation within the plate, the applied load,and the effects of the viscoelastic material properties is exhibitedfor a number of situations, and in each case contrasted withshort- and long-time asymptotic curves based on Tauberian theorems.