Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium

An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. The perturbation technique for solving the nonlinear eigenvalue problem is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law. It is shown that the eigenvalues of the nonlinear eigenvalue problem are fully determined by the exponents of the damage evolution law. In the paper the third-order (four-term) asymptotic expansions of the angular functions determining the stress and continuity fields in the neighborhood of the crack tip are given. The asymptotic expansions of the angular functions permit to find the closed-form solution for the problem considered.     

The Wedge with a Symmetric Crack at the Vertex in Plane Elastostatics

A closed solution of the biharmonic equation under boundaryconditions appropriate to the wedge with a symmetric crack atthe vertex is found by using the Wiener-Hopf technique. Thestress and displacement fields are given as integrals whichmay be readily expressed as eigenfunction expansions. An expression is obtained for the stress intensity factor atthe tip of the crack using an asymptotic method, and is verifiedby a consideration of the stran-energy.     

Stretching of an Elastoplastic Plate with an Arbitrarily Oriented Crack

The problem of determination of the crack resistance of an elastoplastic plate, weakened by a rectilinear slit in the form of a cut, under the conditions of uniaxial stretching is considered. The material of the body is assumed to be incompressible, reinforcing, and obeying the Mises condition of plasticity. The straining theory of plasticity is used. The solutions are obtained in the elastic and plastic regions in the form of asymptotic expansions in the neighborhood of the end of the crack. Based on the conditions that the crack borders are unloaded and the elastic and plastic solutions are conjugate on the contour of the plastic region, unknown constants of integration are found. The two-leafed contours of the plastic region are obtained. The critical load is determined.     

Multi-node higher order expansions of a function

By using the values and higher derivatives of a function at the given nodes, a kind of multi-node higher order expansion of the function is presented. The error terms of the expansions are given. Particular examples are the extensions of the Taylor polynomials, Bernstein polynomials and Lagrange interpolation polynomials. The expansions are numerical approximation polynomials and very useful particular for the functions for which the higher derivatives can be obtained easily.     

椭圆孔边裂纹对SH波的散射及其动应力强度因子

采用复变函数和Green函数方法求解具有任意有限长度的椭圆孔边上的径向裂纹对SH波的散射和裂纹尖端处的动应力强度因子．取含有半椭圆缺口的弹性半空间水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移解作为Green函数,采用裂纹“切割”方法,并根据连续条件建立起问题的定解积分方程,得到动应力强度因子的封闭解答．讨论了孔洞的存在对动应力强度因子的影响．     

Symbolic evaluation of coefficients in Airy-type asymptotic expansions

Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function.     

A novel approach for the computation of orthonormal polynomial expansions

In this paper we present a novel technique for the computation of orthonormal polynomial expansions. The proposed method is very straightforward; given a function to be expanded in a polynomial series, we first use the FFT to compute a vector of Fourier coefficients. Then, using a change of basis transformation, we go from the Fourier coefficients to the polynomial coefficients. Convergence properties for this new approach are investigated.     

具非线性边界条件的Volterra型泛函微分方程边值问题奇摄动

首先利用微分不等式理论和一些分析技巧,探讨了一类具非线性边界条件的二阶Volterra型泛函微分方程边值问题解的存在性问题.然后通过对右端边界层函数和外部解的构造,进一步研究了一类具小参数的二阶Votterra型非线性边值问题.利用微分中值定理和上、下解方法得到了边值问题解的存在性,并给出了解的关于小参数的一致有效渐近展开式.     

夹杂和裂纹的相互作用及端点相交的奇性性态分析

利用单根裂纹和单根夹杂的基本解,通过弹性力学的线性叠加原理,将平面裂纹和夹杂相互作用的问题归结为解一组带有柯西型奇异积分的积分方程组,计算了裂纹和夹杂端点的应力强度因子,给出了一些数值例子,并对夹杂和裂纹水平接触时的情形作了奇性分析,结果可作为研究夹杂尖端引起的裂纹及其扩展的工程分析的计算模型。     

The asymptotic expansion of elastic fields in the vicinity of the contour of a plane crack at the interface of two materials

The stress-strain state in the neighbourhood of the front of a plane crack at the interface of two dissimilar half-spaces of ideally elastic isotropic materials is investigated. The form of the asymptotic expansions of the projections of the displacement vector onto the axis, directed along the tangent, the principal normal and the binormal to the crack contour is obtained. It is shown that asymptotic expansions of the projections of the displacement vector onto directions corresponding to the tangent and principal normal, beginning with the second-order term of the expansion, include both terms with half-integer and complex powers of the distances to the crack contour. This indicates that these projections of the solutions of the three-dimensional problem have singularities, defined by the solutions of both the antiplane and plane strain problems of cracks at the interface of materials. The singularities of the projection of the displacement vector on to the binormal correspond to the singularities of the solution of the plane strain problem.     

二元混合连分式展开的混合差商极限方法

For a univariate function given by its Taylor series expansion,a continuedfraction expansion can be obtained with the Viscovatov's algorithm,as the limitingvalue of a Thiele interpolating continued fraction or by means of the determinantalformulas for inverse and reciprocal differences with coincident data points.In thispaper,both Viscovatov-like algorithms and Taylor-like expansions are incorporatedto yield bivariate blending continued expansions which are computed as the limitingvalue of bivariate blending rational interpolants,which are constructed based on sym-metric blending differences.Numerical examples are given to show the effectivenessof our methods.     

Asymptotic Expansion of Laplace Transforms about the Origin using Generalized Functions

A technique for obtaining asymptotic expansions of integralsby approximating the integrand and then using generalized functiontheory to evaluate the approximated integral is described. Thegeneral technique is presented here by way of a specific example,namely, the expansion of Laplace transforms about the origin.Error bounds for these expansions are also obtained.     

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function ₁ F ₁(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.     

Crack growth modeling via 3D automatic adaptive mesh refinement based on modified-SPR technique

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.     

Multiplier theorems for special Hermite expansions on Cn

The weak type (1,1) estimate for special Hermite expansions on Cn is proved by using the Calderón-Zygmund decomposition. Then the multiplier theorem in Lp(1相似文献   

Continued Fractions which Correspond to Power Series Expansions at Two Points

Theoretical results are derived for constructing continued fractionswhich correspond, in some prescribed way, to given power seriesexpansions at two points. In those cases where the two powerseries are expansions of the same function, new approximationsto the functions are obtained. Error estimates for such rationalfunction approximations are developed in certain cases. Someexamples are given.     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

Some aspects of two-point Padé approximants

Recently McCabe and Murphy have considered the two-point Padé approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.     

The asymptotic solution of the contact problem for a three-dimensional elastic body of finite dimensions

The asymptotic form of Green's vector function with a pole on the boundary is calculated by the method of matched asymptotic expansions. The expansion obtained is used to construct the asymptotic form of the contact pressure. The equations of the contact problem are derived with integral corrections, which take into account the nature of the attachment and the geometry of the elastic body. Examples of calculations for an elliptic punch are given.