一种Trefftz孔洞单元及其在接触问题中的应用

作为一种高效的数值计算工具,Trefftz有限元法愈来愈引入瞩目.与传统有限元法相比,Trefftz有限元法在处理带有孔洞、裂纹、夹杂等局部效应的问题时,无需另外细分网格,只需调整单元域内(Trefftz)插值函数,就能达到比较理想的精度.基于弹性复变理论和保角变换,求出孔洞问题的完备系,以其作为Trefftz插值函数...     

Trefftz有限元法的研究进展

Trefftz有限元法(Trefftz finite element method,TFEM)是一种高效的数值计算方法,兼有传统有限元法和边界元法的诸多优点.基于双独立插值模式,结合杂交泛函和高斯散度定理,推得仅含边界积分的有限元格式.简述了过去10年间(2007—2016)Trefftz有限元法在单元域内插值函数、源项处理、特殊功能单元以及非各向同性材料等方面的研究进展,并对未来的发展趋势给出了几点展望.     

基于光滑边域有限元法的二维裂纹扩展分析

为了提高预测裂纹扩展路径的准确性和效率,本文将光滑边域有限元法和虚拟裂纹闭合法相结合,利用常应变三角形单元,获得裂纹尖端处的断裂控制参量应力强度因子,并运用最大拉应力准则求得裂纹在荷载作用下的启裂方向,对裂纹扩展轨迹给出自动跟踪方法;对三个典型二维裂纹扩展模型,预测了裂纹扩展路径,并将结果与参考文献中的结果进行对比,验证方法的有效性与准确性。数值结果表明:该方法具有单元简单、网格尺寸要求低、裂纹尖端处单元不需特殊处理等优点,是分析裂纹扩展问题简洁高效的数值计算方法。     

基于扩展有限元的应力强度因子的位移外推法

针对平面裂纹问题,阐述了扩展有限元法的单元位移模式、推导了扩展有限元法的控制方程、介绍了特殊单元的数值积分技术.基于最小二乘法,建立了应力强度因子位移外推法的计算公式.利用MATLAB编写计算程序,对平面裂纹问题用扩展有限元法进行了计算.基于扩展有限元法的计算结果,分别利用位移外推法和相互作用积分法,对平面裂纹的应力强度因子进行了计算.计算结果表明,位移外推法比相互作用积分法能更方便和准确地计算平面裂纹的应力强度因子.     

功能梯度板条Ⅲ型运动裂纹问题研究

现存文献关于梯度材料断裂问题的研究大都是假设材料参数为坐标的指数函数或幂函数,而其它函数形式较少采用.本文假设功能梯度材料剪切模量和密度的倒数均为坐标的线性函数,而泊松比为常量,研究功能梯度板条的反平面运动裂纹问题.利用Fourier积分变换技术和传递矩阵法将混合边值问题化为一对奇异积分方程,通过数值求解奇异积分方程获得板条运动裂纹在反平面载荷作用下的动态应力强度因子,并讨论了裂纹运动速度、裂纹相对尺寸、以及材料非均匀性对动态应力强度因子的影响,结果证明梯度参数、裂纹速度和几何尺寸对材料动态断裂行为有显著影响.     

比例边界有限元法求解裂纹面接触问题

比例边界有限元侧面上有任意荷载时,将侧面载荷分解成关于径向方向局部坐标的多项式函数的和,推导给出了考虑侧面载荷存在的新型形函数,并基于该形函数推导了刚度矩阵和等效节点载荷列阵.首次对比例边界有限元法求解裂纹面接触问题进行了研究,运用Lagrange乘子引入接触界面约束条件,推导给出了比例边界有限元求解裂纹面接触问题的控制方程.将裂纹面单元分为非裂尖单元和含有侧面的裂尖单元.在非裂尖单元中的裂纹面,裂纹面作为多边形单元的边界,边界上的接触力可等效到节点上,通过在节点上构造Lagrange乘子,采用点对点接触约束进行处理.对于含有侧面的裂尖单元,在整个侧面上构造Lagrange乘子的插值场,采用边对边接触约束进行处理.对三个不同的接触约束状态下的算例进行了数值计算,通过与解析解及有限元软件ABAQUS计算结果的对比,验证了本文提出的比例边界有限元点对点和边对边接触求解裂纹面接触问题的精确性与有效性.     

CRACK FACE CONTACT PROBLEM ANALYSIS USING THE SCALED BOUNDARY FINITE ELEMENT METHOD

比例边界有限元侧面上有任意荷载时,将侧面载荷分解成关于径向方向局部坐标的多项式函数的和,推导给出了考虑侧面载荷存在的新型形函数,并基于该形函数推导了刚度矩阵和等效节点载荷列阵.首次对比例边界有限元法求解裂纹面接触问题进行了研究,运用Lagrange乘子引入接触界面约束条件,推导给出了比例边界有限元求解裂纹面接触问题的控制方程.将裂纹面单元分为非裂尖单元和含有侧面的裂尖单元.在非裂尖单元中的裂纹面,裂纹面作为多边形单元的边界,边界上的接触力可等效到节点上,通过在节点上构造Lagrange乘子,采用点对点接触约束进行处理.对于含有侧面的裂尖单元,在整个侧面上构造Lagrange乘子的插值场,采用边对边接触约束进行处理.对三个不同的接触约束状态下的算例进行了数值计算,通过与解析解及有限元软件ABAQUS计算结果的对比,验证了本文提出的比例边界有限元点对点和边对边接触求解裂纹面接触问题的精确性与有效性.     

一种基于直接计算高阶奇异积分的断裂力学双边界积分方程分析法

相对于有限元法,边界单元法在求解断裂问题上有着独特的优势,现有的边界单元法中主要有子区域法和双边界积分方程法.采用一种改进的双边界积分方程法求解二维、三维断裂问题的应力强度因子,对非裂纹边界采用传统的位移边界积分方程,只需对裂纹面中的一面采用面力边界积分方程,并以裂纹间断位移为未知量直接用于计算应力强度因子.采用一种高阶奇异积分的直接法计算面力边界积分方程中的超强奇异积分;对于裂纹尖端单元,提供了三种不同形式的间断位移插值函数,采用两点公式计算应力强度因子.给出了多个具体的算例,与现存的精确解或参考解对比,可得到高精度的计算结果.      

分区混合有限元法求混合型应力强度因子

本文在早先工作[1],[2]的基础上,进一步用分区混合有限元法求解平面断裂问题的应力强度因子,作了两点改进:1.余能区单元采用多个应力参数;2.求解的问题可包括混合型问题。文中给出了几个典型算例,显示出本法的一些优点。     

分区混合有限元法求混合型应力强度因子

本文在早先工作[1],[2]的基础上,进一步用分区混合有限元法求解平面断裂问题的应力强度因子,作了两点改进:1.余能区单元采用多个应力参数;2.求解的问题可包括混合型问题。文中给出了几个典型算例,显示出本法的一些优点。     

Mode III fracture analysis by Trefftz boundary element method

This paper presents a hybrid Trefftz (HT) boundary element method (BEM) by using two indirect techniques for mode III fracture problems. Two Trefftz complete functions of Laplace equation for normal elements and a special purpose Trefftz function for crack elements are proposed in deriving the Galerkin and the collocation techniques of HT BEM. Then two auxiliary functions are introduced to improve the accuracy of the displacement field near the crack tips, and stress intensity factor (SIF) is evaluated by local crack elements as well. Furthermore, numerical examples are given, including comparisons of the present results with the analytical solution and the other numerical methods, to demonstrate the efficiency for different boundary conditions and to illustrate the convergence influenced by several parameters. It shows that HT BEM by using the Galerkin and the collocation techniques is effective for mode III fracture problems. The project supported by the National Natural Science Foundation of China(10472082). The English text was polished by Keren Wang.     

APPLICATION OF TREFFTZ BEM TO ANTI-PLANE PIEZOELECTRIC PROBLEM

Anti-plane electroelastic problems are studied by the Trefftz boundary elementmethod (BEM) in this paper. The Trefftz BEM is based on a weighted residual formulation andindirect boundary approach. In particular the point-collocation and Galerkin techniques, in whichthe basic unknowns are the retained expansion coefficients in the system of complete equations,are considered, Furthermore, special Trefftz functions and auxiliary functions which satisfy ex-actly the specified boundary conditions along the slit boundaries are also used to derive a specialpurpose element with local defects. The path-independent integral is evaluated at the tip of acrack to determine the energy release rate for a mode Ⅲ fracture problem. In final, the accuracyand efficiency of the Trefftz boundary element method are illustrated by an example and thecomparison is made with other methods.     

Introducing new cracked finite elements and a method for SIF calculation of cracks

Two different types of 8-node cracked quadrilateral finite element are presented for fracture applications. The first element contains a central crack and the other one includes an edge crack. The introduced elements are applicable in 2D problems. The crack is not physically modeled within the element, but instead, its effects on the stiffness matrix are taken into account by utilizing linear fracture mechanics laws. Furthermore, a simple and practical procedure is proposed for calculation of stress intensity factor (SIF) by employing proposed cracked elements. Several numerical examples are presented to evaluate the capabilities of the proposed elements and procedure.     

有限厚度板穿透裂纹前缘附近三维弹性应力场分析

通过三维有限元计算来研究有限宽度、有限厚度含有穿透裂纹板的裂纹前缘应力场，从中找出应力强度因子与板的厚度、裂纹长度之间的关系，同时还分析了裂尖的三维约束程度和三维约束区的大小。分析结果表明：应力强度因子沿厚度的分布是不均匀的，应力强度因子的最大值及其位置与厚度有关；有限厚度板中面应力强度因子(KI)m-p及最大应力强度因子(KI)max均大于平面应力或平面应变的应力强度因子。对有限厚度裂纹问题，按平面应力或平面应变来考虑是不安全的；板中面的应力强度因子(KI)m-p及最大应力强度因子(KI)max是厚度B／a的函数；板的中面离面约束系数Tx最大，自由面(z=B)Tx=0。沿厚度方向裂尖附近的离面约束系数Tx也是z／B和B／a的函数，随着厚度的增加离面约束系数Tx增大，离中面越近离面约束系数Tx越大。Tx随着x的增大急剧减小，三维约束影响区域大小大约为板厚的一半，且裂纹长度a／W对应力强度因子沿厚度变化规律及Tx影响区域大小影响较小。     

表面裂纹问题瞬态响应的边界元分析

本文将拉氏变换-边界元法用于表面裂纹问题的瞬态响应分析。文中讨论了拉氏反演参数的选择和动态应力强度因子的计算方法。作为程序的考核和离散方案的选择,分别地计算了水平柱体一端固定、另一端受p(t)=poH(t)载荷时的位移响应和具有贯穿裂纹的厚板在两种离散方案时的动态应力强度因子响应。最后,还计算了若干载荷工况的半圆表面裂纹板应力强度因子的瞬态响应,获得了有效的数值结果。     

Hybrid finite element formulations for elastodynamic analysis in the frequency domain

Three alternative sets of hybrid formulations to solve linear elastodynamic problems by the finite element method are presented. They are termed hybrid–mixed, hybrid and hybrid–Trefftz and differ essentially on the field conditions that the approximation functions are constrained to satisfy locally. Two models, namely the displacement and the stress models, are obtained for each formulation depending on whether the tractions or the boundary displacements are the field chosen to implement interelement continuity. A Fourier time discretization is used to uncouple the solving system in the frequency domain. The basic space discretization criterion is implemented directly on the fundamental relations of elastodynamics and used to derive the stress and displacement models of the hybrid–mixed formulation. The hybrid and hybrid–Trefftz formulations are presented in sequence as the variants of the hybrid–mixed formulation obtained by progressively increasing the constraints on the approximation bases. Numerical implementation aspects are briefly discussed and the performance of the finite element models is illustrated with numerical applications.     

On the uses of special crack tip elements in numerical rock fracture mechanics

Numerical methods such as boundary element methods are widely used for the stress analysis in solid mechanics. These methods are also used for crack analysis in rock fracture mechanics. There are singularities for the stresses and displacements at the crack tips in fracture mechanics problem, which decrease the accuracy of the numerical results in areas very close to the crack ends. To overcome this, higher order elements and isoperimetric higher order elements have been used. Recently, special crack tip elements have been proposed and used in most of the numerical fracture mechanics models. These elements can drastically increase the accuracy of the results near the crack tips, but in most of the models only one special crack tip element has been used for each crack end. In this study the uses of higher order crack tip elements are discussed and a higher order displacement discontinuity method is used to investigate the effect of these elements on the accuracy of the results in some crack problems. The useful shape functions for two special crack tip elements, are derived and given in the text and appendix for both infinite and semi-infinite plane problems. In this analysis both Mode I and Mode II stress intensity factors are computed . Some example problems are solved and the computed results are compared with the results given in the literature. The numerical results obtained here are in good agreement with those cited in the literature. For the curved crack problem, the strain energy release rate, G can be calculated accurately in the vicinity of the crack tips by using the higher order displacement discontinuity method with a quadratic variation of displacement discontinuity elements and with two special crack tip elements at each crack end.     

Behaviors of a single crack in multiple bolted joints

This paper examines the pin load ratios and the stress intensity factors (SIFs) of a single crack in the multiple bolted joints by using finite element analyses. Cubic-spline contact elements and rigid links were used to model the contact surface between the bolt and the rigid pin. The least-squares method was used to determine the SIFs. The finite element results indicate that the cracked hole can still sustain the major part of the original loading at the uncracked condition. The first hole sustains the largest pin load and mode-I SIF, which are reduced little for crack propagation. This critical condition cannot be reduced by the arrangement of more pins in the plate. In this paper, two simple formulae were also investigated to fit the load ratios and SIFs of the multiple bolted-joints problems.     

Kinked crack analysis by a hybridized boundary element/boundary collocation method

In this research a two dimensional displacement discontinuity method (which is a kind of indirect boundary element method) using higher order elements (i.e. a source element with a cubic variation of displacement discontinuities having four sub-elements) is used to obtain the displacement discontinuities along each boundary element. In this paper, three kinds of the higher order boundary elements are used: the ordinary elements, the kink elements and the special crack tip elements.The boundary collocation technique is used for the calculation of the displacement discontinuities at the center of each sub-elements. Again a special boundary collocation technique is used to treat the kinked source elements occur in the crack analysis. Considering the two source elements (each having four sub-elements) joined at a corner (kink point). The collocation points in the cubic element model which are outside of the kink point are moved to the crack kink then the displacement discontinuities on the left and right sides of the kink are calculated. The displacement discontinuities of the kink point are obtained by averaging the corresponding values of its left and right sides. The special crack tip elements are also treated by the boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip. Some simple example problems are solved numerically by the proposed method. The numerical results are compared with the corresponding results obtained by the previous methods cited in the literature. This comparison shows a very good agreement between the results and verify the accuracy and validity of the proposed method.     

一种XFEM断裂分析的裂尖单元新型改进函数

提出了一种适用于裂尖改进单元的新型改进函数, 基于三角变换的方法, 保留裂纹尖端场的应力奇异性和裂纹上、下表面的位移不连续性, 将常规扩展有限元法裂尖改进单元的4 项改进函数缩减为2 项, 裂尖改进单元的结点由常规的8 个改进自由度减少为4 个. 采用2 个正交的水平集函数表征材料内部裂纹面, 详细阐述了改进单元类型的判别方法, 给出一种改进单元的分区域积分方案. 最后, 若干断裂力学问题经典算例的数值计算结果表明:建议的裂尖改进函数具有较高的数值精度, 该方法是十分有效的.