正交异性光弹性应力分析方法研究

本文建立了正交异性光弹性应力分析的实验边界元混合解法并编制了相应的程序系统BE-MSC.该方法仅需模型边界结点处等差线一项实验数据即可实施应力的分离。它还十分适用于存在初应力的模型材料光弹性分析问题。最后对正交异性对径受压圆盘进行了光弹性应力实例分析。      

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

Complex boundary integral equations (Fredholm‐type regular or Cauchy‐type singular or even Hadamard–Mangler‐type hypersingular) have been used for the numerical solution of general plane isotropic elasticity problems. The related Muskhelishvili and, particularly, Lauricella–Sherman equations are famous in the literature, but several more extensions of the Lauricella–Sherman equations have also been proposed. In this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries (for anyone of the above equations: regular or singular or hypersingular, but with a prerequisite their actual numerical solution) through the appropriate use of the even more classical elementary Cauchy theorem in complex analysis. This approach has been already used for the accurate numerical computation of analytic functions and their derivatives by Ioakimidis, Papadakis and Perdios (BIT 1991; 31 : 276–285), without applications to elasticity problems, but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non‐analytic complex density function (such as an edge dislocation/loading distribution density) is numerically available on the boundary. The present results are also directly applicable to inclusion problems, anisotropic elasticity, antiplane elasticity and classical two‐dimensional fluid dynamics, but, unfortunately, not to crack problems in fracture mechanics. Brief numerical results (for the complex potentials), showing the dramatic increase of the computational accuracy, are also displayed and few generalizations proposed. Copyright © 2000 John Wiley & Sons, Ltd.     

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.     

求解夹杂问题的有限部积分──边界元法

利用Kelvin解及有限部积分的概念和方法,导出求解含夹杂二维有限弹性体的超奇异积分方程,继而使用有限部积分与边界元结合的方法,为其建立了数值求解方法,即有限部积分与边界元法.最后计算了若干典型数值例子夹杂端部的应力强度因子.      

Tangential derivative of singular boundary integrals with respect to the position of collocation points

This paper investigates the evaluation of the sensitivity, with respect to tangential perturbations of the singular point, of boundary integrals having either weak or strong singularity. Both scalar potential and elastic problems are considered. A proper definition of the derivative of a strongly singular integral with respect to singular point perturbations should accommodate the concomitant perturbation of the vanishing exclusion neighbourhood involved in the limiting process used in the definition of the integral itself. This is done here by esorting to a shape sensitivity approach, considering a particular class of infinitesimal domain perturbations that ‘move’ individual points, and especially the singular point, but leave the initial domain globally unchanged. This somewhat indirect strategy provides a proper mathematical setting for the analysis. Moreover, the resulting sensitivity expressions apply to arbitrary potential-type integrals with densities only subjected to some regularity requirements at the singular point, and thus are applicable to approximate as well as exact BEM solutions. Quite remarkable is the fact that the analysis is applicable when the singular point is located on an edge and simply continuous elements are used. The hypersingular BIE residual function is found to be equal to the derivative of the strongly singular BIE residual when the same values of the boundary variables are substituted in both SBIE and HBIE formulations, with interesting consequences for some error indicator computation strategies. © 1998 John Wiley & Sons, Ltd.     

Use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicators

In this work, a new global reanalysis technique for the efficient computation of stresses and error indicators in two‐dimensional elastostatic problems is presented. In the context of the boundary element method, the global reanalysis technique can be viewed as a post‐processing activity that is carried out once an analysis using Lagrangian elements has been performed. To do the reanalysis, the functional representation for the displacements is changed from Lagrangian to Hermite, introducing the nodal values of the tangential derivatives of those quantities as additional degrees of freedom. Next, assuming that the nodal values of the displacements and the tractions remain practically unchanged from the ones obtained in the analysis using Lagrangian elements, the tangent derivative boundary integral equations are collocated at each functional node in order to determine the additional degrees of freedom that were introduced. Under this scheme, a second system of equations is generated and, once it is solved, the nodal values of the tangential derivatives of the displacements are obtained. This approach gives more accurate results for the stresses at the nodes since it avoids the need to differentiate the shape functions in order to obtain the normal strain in the tangential direction. When compared with the use of Hermite elements, the global reanalysis technique has the attraction that the user does not have to give as input data the additional information required by this type of elements. Another important feature of the proposed approach is that an efficient error indicator for the values of the stresses can also be obtained comparing the values for the stresses obtained through the use of Lagrangian elements and the global reanalysis technique. Copyright © 2001 John Wiley & Sons, Ltd.     

EVALUATION OF THE STRESS TENSOR IN 3-D ELASTOPLASTICITY BY DIRECT SOLVING OF HYPERSINGULAR INTEGRALS

A 3-D hypersingular Boundary Integral Equation (BIE) of elastoplasticity is derived. Using this formulation the displacement rate gradients and the complete stress tensor on the boundary can be evaluated directly as opposed to the classical approach, where the shape functions derivatives are to be calculated. The regularization of strongly singular and hypersingular boundary integrals, as well as strongly singular domain integrals for a source point positioned on the boundary is carried out in a general manner. Arbitrary types of elements and arbitrary positions of the source point with respect to continuity requirements can be used. Numerical 3-D elastoplastic examples (notch and crack problems) illustrate the advantages of the proposed method.     

A boundary spectral method for solving exterior acoustical problems with hypersingular integrals

A boundary spectral method is developed to solve acoustical problems with arbitrary boundary conditions. A formulation, originally derived by Burton and Miller, is used to overcome the non‐uniqueness problem in the high wave number range. This formulation is further modified into a globally non‐singular form to simplify the procedure of numerical quadrature when spectral methods are applied. In the present approach, generalized Fourier coefficients are determined instead of local variables at nodes as in conventional methods. The convergence of solutions is estimated through the decay of magnitude of the generalized Fourier coefficients. Several scattering and radiation problems from a sphere are demonstrated with high wave numbers in the present paper. Copyright © 1999 John Wiey & Sons, Ltd.     

BOUNDARY ELEMENT ANALYSIS OF STRESS CONCENTRATION IN MICROPOLAR ELASTIC PLATE

Based on Eringen's Micropolar Elasticity Theory, a two-dimensional Boundary Element (BEM) formulation is derived and the corresponding computer programs are developed to solve the two-dimensional stress concentration problems of micropolar elastic material with arbitrary non-smooth surfaces. The new 2-D BEM formulation is first applied to solve the stress concentration around a hole in a plate under tension, and shown to be in excellent agreement with existing analytical solutions. Next, the new BEM formulation is applied to solve, for the first time, the stress concentration around a semicircular edge notch in a plate under tension. The effects of coupling number, characteristic length and radius/width ratio on the micropolar effects are derived, and the important new findings are summarized. © 1997 by John Wiley & Sons, Ltd.     

Flux and traction boundary elements without hypersingular or strongly singular integrals

The present paper deals with a boundary element formulation based on the traction elasticity boundary integral equation (potential derivative for Laplace's problem). The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require any change of co‐ordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. The formulation presented is completely general and valid for arbitrary shaped open or closed boundaries. Analytical expressions for all the required hypersingular or strongly singular integrals are given in the paper. To fulfil the continuity requirement over the primary density a simple BE discretization strategy is adopted. Continuous elements are used whereas the collocation points are shifted towards the interior of the elements. This paper pretends to contribute to the transformation of hypersingular boundary element formulations as something clear, general and easy to handle similar to in the classical formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

HYPERSINGULAR BEM FOR TRANSIENT ELASTODYNAMICS

The topic of hypersingular boundary integral equations is a rapidly developing one due to the advantages which this kind of formulation offers compared to the standard boundary integral one. In this paper the hypersingular formulation is developed for time-domain antiplane elastodynamic problems. Firstly, the gradient representation is found from the displacement one, removing the strong singularities (Dirac's delta functions) which arise due to the differentiation process. The gradient representation is carried to the boundary through a limiting process and the resulting equation is shown to be consistent with the static formulation. Next, the numerical treatment of the traction boundary integral equation and its application to crack problems are presented. For the boundary discretization, conforming quadratic elements are tested, which are introduced in this paper for the first time, and it is shown that the results are very good in spite of the lesser number of unknowns of this approach in comparison to the non-conforming element alternative. A procedure is devised to numerically perform the hypersingular integrals that is both accurate and versatile. Several crack problems are solved to show the possibilities of the method. To this end both straight and curved elements are employed as well as regular and distorted quarter point elements.     

Numerical solution of the variation boundary integral equation for inverse problems

In this paper a procedure to solve the identification inverse problems for two‐dimensional potential fields is presented. The procedure relies on a boundary integral equation (BIE) for the variations of the potential, flux, and geometry. This equation is a linearization of the regular BIE for small changes in the geometry. The aim in the identification inverse problems is to find an unknown part of the boundary of the domain, usually an internal flaw, using experimental measurements as additional information. In this paper this problem is solved without resorting to a minimization of a functional, but by an iterative algorithm which alternately solves the regular BIE and the variation BIE. The variation of the geometry of the flaw is modelled by a virtual strainfield, which allows for greater flexibility in the shape of the assumed flaw. Several numerical examples demonstrate the effectiveness and reliability of the proposed approach. Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals

The typical Boundary Element Method (BEM) for fourth‐order problems, like bending of thin elastic plates, is based on two coupled boundary integral equations, one strongly singular and the other hypersingular. In this paper all singular integrals are evaluated directly, extending a general method formerly proposed for second‐order problems. Actually, the direct method for the evaluation of singular integrals is completely revised and presented in an alternative way. All aspects are dealt with in detail and full generality, including the evaluation of free‐term coefficients. Numerical tests and comparisons with other regularization techniques show that the direct evaluation of singular integrals is easy to implement and leads to very accurate results. Copyright © 1999 John Wiley & Sons, Ltd.     

On the Cauchy principal value of the surface integral in the boundary integral equation of 3D elasticity

A simple demonstration of the existence of the Cauchy principal value (CPV) of the strongly singular surface integral in the Somigliana Identity at a non-smooth boundary point is presented. First a regularization of the strongly singular integral by analytical integration of the singular term in the radial direction in pre-image planes of smooth surface patches is carried out. Then it is shown that the sum of the angular integrals of the characteristic of the tractions of the Kelvin fundamental solution is zero, a formula for the transformation of angles between the tangent plane of a suface patch and the pre-image plane at smooth mapping of the surface patch being derived for this purpose.     

Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation

Discretization of boundary integral equations leads, in general, to fully populated complex valued non-Hermitian systems of equations. In this paper we consider the efficient solution of these boundary element systems by preconditioned iterative methods of Krylov subspace type. We devise preconditioners based on the splitting of the boundary integral operators into smooth and non-smooth parts and show these to be extremely efficient. The methods are applied to the boundary element solution of the Burton and Miller formulation of the exterior Helmholtz problem which includes the derivative of the double layer Helmholtz potential—a hypersingular operator. © 1998 John Wiley & Sons, Ltd.     

弹性力学辛体系广义动量矩定理

类似于动力学中动量矩的概念,提出了弹性力学辛体系中广义动量矩的概念,给出了平面直角坐标辛体系和极坐标辛体系中广义动量、广义动量矩的统一定义,对广义动量矩的相关理论进行了研究。由Hamilton对偶方程导出了广义动量矩定理,得到了广义动量矩的守恒律,给出了守恒条件,分析了广义动量矩定理和守恒律的物理意义,经典问题验证了广义动量矩定理和守恒律。这一定理进一步丰富了弹性力学辛体系理论的内容,它和Hamilton函数、广义动量等相关理论一起,形成了辛体系中类似于动力学基本定理的一个理论体系。     

AN INDIRECT BOUNDARY ELEMENT METHOD FOR PLATE BENDING ANALYSIS

An indirect Boundary Element Method is employed for the static analysis of homogeneous isotropic and linear elastic Kirchhoff plates of an arbitrary geometry. The objectives of this paper consists of a construction and a study of the resulting boundary integral equations as well as a development of stable powerful algorithms for their numerical approximation. These equations involve integrals with high-order kernel singularities. The treatment of singular and hypersingular integrals and a construction of solutions in the neighborhood of the irregular points on the boundary are discussed. Numerical examples illustrate the procedure and demonstrate its advantages. © 1997 by John Wiley & Sons, Ltd.     

Weakly singular stress‐BEM for 2D elastostatics

A weakly singular stress‐BEM is presented in which the linear state regularizing field is extended over the entire surface. The algorithm employs standard conforming C⁰ elements with Lagrangian interpolations and exclusively uses Gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space. The linear state stress‐BIE on which the algorithm is based has no free term so that the BEM treatment of external corners requires no special consideration other than to admit traction discontinuities. The self‐regularizing nature of the Somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary. A key observation is the relation between BIE density functions and successful interpolation orders. Numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations. Additionally, the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions. Copyright © 1999 John Wiley & Sons, Ltd.     

层内混杂复合材料断裂纤维附近的应力重分布

研究了层内混杂复合材料的高模量纤维断裂引起的邻近各层应力重分布问题。通过建立适当的计算模型,利用二维弹性力学精确解和付氏变换法,建立问题的奇异积分方程组,通过求解方程组,计算高、低模量层和基体层的应力集中因子,计算结果对"混杂效应"作出了理论解释,并与一般采用的Shear-Lag理论计算结果作了比较,本文结果更精确、合理,可应用于层内混杂复合材料的设计。     

A Galerkin boundary integral method for multiple circular elastic inclusions

The problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd.