压电材料反平面切口奇性指数分析

为理解压电材料反平面切口尖端奇异状态,提出了一种切口奇性特征分析法.基于切口根部位移场幂级数渐近展开假设,从应力平衡方程和电荷守恒条件出发,导出了关于压电材料反平面切口奇性指数的特征微分方程组,并将切口的力电学边界条件以及界面协调条件表达为奇性指数和特征角函数的组合.从而,压电材料切口反平面奇性指数的计算被转化为相应边界条件下常微分方程组特征值的求解问题,采用插值矩阵法可以计算出各阶奇性指数和相应的特征角函数.该法既适合裂纹奇性分析,也可用于单、双材料切口的奇性计算,并避免了用迭代法求解超越方程的不足.因而具有适应性强的特点.计算发现,压电材料反平面切口存在两个奇性指数,切口的奇异性程度随着开角的增大而增强.     

角度非均匀材料平面V形切口应力奇性分析

提出了一种确定角度非均匀材料平面V形切口尖端应力奇性指数的有效方法。首先,在弹性力学基本方程中引入V形切口尖端位移场的级数渐近展开,建立以位移为特征函数的变系数和非线性微分方程组。然后,采用微分求积法(DQM)求解微分方程组,可得到多阶应力奇性指数及其相对应的特征函数,该法具有公式简单、编程方便、计算量少和精度高等优点,可处理任意开口角度和任意材料组合的V形切口。典型算例验证了微分求积法的有效性和精确性。     

V形切口应力强度因子的一种边界元分析方法

将V形切口结构分成围绕切口尖端的小扇形和剩余结构两部分. 尖端处扇形域应力场表示成关于尖端距离$\rho$的渐近级数展开式,从线弹性理论方程推导出了一组分析平面V形切口奇异性的常微分方程特征值问题,通过求解特征方程,得到前若干个奇性指数和相应的特征向量. 再将切口尖端的位移和应力表示为有限个奇性阶和特征向量的组合. 然后用边界元法分析挖去小扇形后的剩余结构. 将位移和应力的线性组合与边界积分方程联立,求解获得切口根部区域的应力场、应力幅值系数和整体结构的位移和应力. 从而准确计算出平面V形切口的奇异应力场和应力强度因子.      

边界元法计算切口多重应力奇性指数

提出采用边界元法直接计算V形切口的多重应力奇性指数。首先在切口尖端挖出一微小扇形域,在该域边界列常规边界积分方程,后将扇形域内的位移场和应力场表示成关于切口尖端距离ρ的渐近级数展开式,回代入切口边界积分方程,离散后得到关于切口奇性指数的代数特征方程,从而求解获得V形切口的应力奇性指数。该法避免了常规边界元法和有限元法在切口尖端附近布置细密单元的缺陷,并可同时求得多阶应力奇性指数。     

一种新的计算各向异性材料裂纹尖端应力强度因子杂交元法

利用有限元特征分析法研究了平面各向异性材料裂纹端部的奇性应力指数以及应力场和位移场的角分布函数,以此构造了一个新的裂纹尖端单元。文中利用该单元建立了研究裂纹尖端奇性场的杂交应力模型,并结合Hellinger-Reissner变分原理导出应力杂交元方程,建立了求解平面各向异性材料裂纹尖端问题的杂交元计算模型。与四节点单元相结合,由此提出了一种新的求解应力强度因子的杂交元法。最后给出了在平面应力和平面应变下求解裂纹尖端奇性场的算例。算例表明,本文所述方法不仅精度高,而且适应性强。     

插值矩阵法分析与复合材料界面相交的平面裂纹奇异应力场

提出了用插值矩阵法分析与各向异性材料界面相交的平面裂纹应力奇异性。基于V形切口尖端附近区域位移场渐近展开,将位移场的渐近展开式的典型项代入线弹性力学基本方程,得到关于平面内与复合材料界面相交的裂纹应力奇异性指数的一组非线性常微分方程的特征值问题,运用插值矩阵法求解,获得了平面内各向异性结合材料中与界面以任意角相交的裂纹尖端的应力奇异性指数随裂纹角的变化规律,数值计算结果与已有结果比较表明,本文方法具有很高的精度和效率。     

三维切口应力奇性指数计算

将三维切口根部的位移渐近展开式引入线弹性力学平衡方程,导得关于切口应力奇性指数的特征微分方程组。再采用插值矩阵法,一次性地计算出三维切口的各阶应力奇性指数,它们具有同阶精度,并可同时获取相应的特征角函数。算例显示该法是分析三维切口应力奇异指数的一个有效的路径。计算结果表明,若直接用平面应变理论预测三维切口应力奇性指数将导致部分重要的奇性指数丢失。     

各向异性复合材料尖劈和接头的奇性应力指数研究

提出了一个新的、基于位移的、求解三维尖劈端部奇性应力指数问题的非协调元特征分析法。该方法假定尖劈端部邻域内的位移场没有采用奇异变换技术，导出虚功方程的出发点不同于过去原有求解裂纹尖端近似场的有限元特征分析法，在有限元离散时采用的单元形式为非协调元。文中运用该方法给出了若干求解各向异性复合材料尖劈／接头端部奇性应力指数的算例。所有的计算结果表明，本文方法能够求解复杂尖劈／接头的全部奇性应力指数，使用的单元少而且精度高。     

复合材料Reissner板中与界面相交的裂纹奇异性分析

基于切口尖端附近区域位移场的渐近展开,提出了分析复合材料板中与界面相交的切口应力奇异性的新方法。将位移场渐近展开式的典型项代入弹性板的基本方程,得到关于复合材料板中与界面相交的切口应力奇异性指数的一组非线性常微分方程的特征值;采用变量代换法,将非线性特征问题转化为线性特征问题,并用插值矩阵法求解获得了各向异性结合材料中与界面以任意角相交的裂纹尖端的应力奇异性指数随裂纹角的变化规律;最后将计算结果与现有结果进行对比。结果表明:两种结果吻合较好,表明本文方法是有效的。     

与界面垂直相交V形切口的边界元分析方法

建立了边界元法计算各向同性结合材料中与界面垂直相交V形切口奇异应力场的分析方法。首先将V形切口尖端附近区域的位移场和应力场用Williams渐近展开式表达,将其代入弹性力学基本方程中,由插值矩阵法获得应力奇异性指数及其对应的位移函数;然后在V形切口尖端区域挖取一个小扇形域,将该扇形区域的位移场表示为有限项奇性指数和特征角函数的线性组合,并对挖去小扇形域后的剩余结构建立边界积分方程;最后将扇形区域位移场表达式和边界积分方程联合求出其切口尖端位移场的组合系数,从而获得各向同性结合材料中与界面垂直相交V形切口尖端的应力强度因子。本文的计算结果与现有结果对比吻合良好,表明了本文方法的有效性。     

反平面V形切口塑性应力奇异性分析

论文提出了用插值矩阵法计算幂硬化塑性材料反平面V形切口和裂纹尖端区域的应力奇异性.首先在切口和裂纹尖端区域采用自尖端径向度量的渐近位移场假设,将其代入塑性全量理论的基本微分方程后,推导出包含应力奇异性特征指数和特征角函数的非线性常微分方程特征值问题.然后采用插值矩阵法迭代求解导出的控制方程,得到一般的塑性材料反平面V形切口和裂纹的前若干阶应力奇异阶和相应的特征角函数,该法的重要优点是以上求解的特征角函数和它们各阶导函数具有同阶精度,并且一次性地求出前若干阶特征对.同时,插值矩阵法计算量小,易于和其他方法联合使用,这些优点在后续求解尖端区域完全应力场非常优越.论文方法的计算结果与现有结果对照,发现吻合良好,表明了论文方法的有效性.     

插值矩阵法分析双材料平面V形切口奇异阶

对二维V形切口问题提出奇异阶分析的一个新方法.首先,以V形切口尖端附近位移场沿其径向渐近展开为基础,将其线弹性理论控制方程转换成切口尖端附近关于周向变量的常微分方程组特征值问题,然后将数值求解两点边值问题的插值矩阵法进一步拓展为求解一般常微分方程组特征值问题,插值矩阵法是在离散节点上采用微分方程中待求函数的最高阶导数作为基本未知量.由此,V形切口的应力奇性阶问题通过插值矩阵法获得,同时相应的切口附近位移场和应力场特征向量一并求出.     

A new boundary element approach of modeling singular stress fields of plane V-notch problems

In this paper, a new boundary element (BE) approach is proposed to determine the singular stress field in plane V-notch structures. The method is based on an asymptotic expansion of the stresses in a small region around a notch tip and application of the conventional BE in the remaining region of the structure. The evaluation of stress singularities at a notch tip is transformed into an eigenvalue problem of ordinary differential equations that is solved by the interpolating matrix method in order to obtain singularity orders (degrees) and associated eigen-functions of the V-notch. The combination of the eigen-analysis for the small region and the conventional BE analysis for the remaining part of the structure results in both the singular stress field near the notch tip and the notch stress intensity factors (SIFs).Examples are given for V-notch plates made of isotropic materials. Comparisons and parametric studies on stresses and notch SIFs are carried out for various V-notch plates. The studies show that the new approach is accurate and effective in simulating singular stress fields in V-notch/crack structures.     

Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions

The extended Kantorovich method is employed to study the local stress concentrations at the vicinity of free edges in symmetrically layered composite laminates subjected to uniaxial tensile load upon polynomial stress functions. The stress fields are initially assumed by means of the Lekhnitskii stress functions under the plane strain state. Applying the principle of complementary virtual work, the coupled ordinary differential equations are obtained in which the solutions can be obtained by solving a generalized eigenvalue problem. Then an iterative procedure is estab-lished to achieve convergent stress distributions. It should be noted that the stress function based extended Kantorovich method can satisfy both the traction-free and free edge stress boundary conditions during the iterative processes. The stress components near the free edges and in the interior regions are calculated and compared with those obtained results by finite element method (FEM). The convergent stresses have good agreements with those results obtained by three dimensional (3D) FEM. For generality, various layup configurations are considered for the numerical analysis. The results show that the proposed polynomial stress function based extended Kan-torovich method is accurate and efficient in predicting the local stresses in composite laminates and computationally much more efficient than the 3D FEM.     

On the asymptotics of the stress and strain fields near a crack tip

The problem is solved under the plane strain conditions for a crack of general form, which in general is neither a mode I nor a mode II crack. We assume that the strains are small and the material is nonlinearly elastic. The mathematical statement of the problem is reduced to the eigenvalue problem for a system of ordinary nonlinear differential equations. Its solution is obtained numerically. We show that, for an incompressible material with power-law relations between the stress and strain deviators, the solution (the well-known HRR-asymptotics [1, 2]) exists only for mode I and II cracks. In the general case, we can only speak of approximate solutions. A similar conclusion can be made for different-modulus materials. We analyze the results of the preceding papers [1–7], where specific cases of the problem were considered.     

COMPUTATION OF STRESS INTENSITY FACTORS BY THE SUB-REGION MIXED FINITE ELEMENT METHOD OF LINES

Based on the sub-region generalized variational principle,a sub-region mixed ver- sion of the newly-developed semi-analytical‘finite element method of lines’(FEMOL)is pro- posed in this paper for accurate and efficient computation of stress intensity factors(SIFs)of two-dimensional notches/cracks.The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used,with the sought SIFs being among the unknown coefficients.The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements.A mixed system of ordinary differential equations(ODEs) and al- gebraic equations is derived via the sub-region generalized variational principle.A singularity removal technique that eliminates the stress parameters from the mixed equation system even- tually yields a standard FEMOL ODE system,the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver.A number of numerical examples,including bi-material notches/cracks in anti-plane and plane elasticity,are given to show the generally excellent performance of the proposed method.