复合材料闭口薄壁杆件的平衡问题

本文根据现有均匀各向同性材料薄壁杆件的理论,导出了复合材料闭口薄壁杆件拉伸、弯曲、自由扭转和约束扭转的基本方程和有关的公式,可供工程设计计算之用。      

组合断面薄壁杆件弯扭耦合分析

根据薄壁杆件结构约束扭转的一致性理论,研究了由多个薄壁杆件组成的组合薄壁杆件结构的弯扭耦合问题。在符拉索夫刚周边假定,库尔布鲁纳-哈丁理论对纵向翘曲位移的假定和弯曲时的平截面假定下,得到了弯扭耦合作用下组合断面薄壁杆件结构的总势能,并由此得出相应的拉格朗日函数。引入对偶变量,建立了组合断面薄壁杆件结构静力分析的哈密顿对偶体系,导出了弯扭耦合分析的哈密顿正则方程。用两端边值问题的精细积分法可求出高精度数值解。这种方法适合于开口断面、闭口断面及开闭口混合断面薄壁杆件结构的弯扭耦合分析。该方法是哈密顿力学在组合断面薄壁杆件结构弯扭耦合分析中的应用,数学推导过程简单,且有成熟高效的数值算法,思路清晰、精度高、易于接受。     

薄壁杆件弯曲和扭转耦合动力方程

本文综合考虑梁的弯曲剪切变形和薄壁杆件的约束扭转变形，由能量原理虚功方程的变分形式。     

结构中裂纹杆件的动力学分析

杆系结构是一类应用广泛的工程结构，作为杆系结构的基本元素－杆件，常常受到拉、压、弯、扭的联合作用。裂纹杆件的扭转和弯曲，是现代工程结构中一类重要的问题。对于横截面带有裂纹的杆系结构的力学分析，文献上报道较少。本文在先前工作的基础上，进一步提出弯扭庆力函数的概念，将裂纹杆件的扭转问题、弯曲问题和弯扭复合加载问题纳入统一的框架之下进行讨论，得到了用弯扭应力函数表示的剪应力通用表达式。以偏心裂纹矩形截面杆件的扭转问题为例，利用单裂纹一般解及裂纹分割法，导出了问题一组广义柯西型奇异积分方程，通过数值求解，计算了裂纹杆件的扭转刚度以及裂纹尖端的应力强度因子。     

基于协调翘曲场的开闭口混合薄壁截面杆件约束扭转一维有限元分析

开口及闭口薄壁杆件约束扭转问题已由经典Timoshenko和Benscoter理论解决。然而,开闭口混合薄壁截面杆件约束扭转分析必须考虑开、闭口部分翘曲能力的差异,翘曲剪流形成机理有待进一步研究。该文假定开、闭口截面翘曲分别满足Vlasov和Umanskii假定,考虑开、闭口截面公共节点翘曲连续性要求,建立含有待定翘曲参数的协调翘曲模型。由截面受力平衡,确定翘曲参数显式列式,提出开闭口混合薄壁截面杆件约束扭转分析的一维有限元模型。算例及参数分析结果表明,基于Umanskii第二理论的Ⅰ类方法在悬臂板及闭口周边引入附加剪流,影响翘曲剪应力精度。基于Umanskii第二理论的Ⅱ类方法只能计算截面板件平均剪应力,无法反映真实翘曲剪流分布。基于Vlasov约束扭转假定的Beam-189单元忽略闭口周边约束效应产生的附加翘曲及剪流,影响翘曲正应力和剪应力精度。该文方法与Shell-63单元能得到基本吻合的变形与应力结果,说明一维梁元能正确反映开闭口混合薄壁截面杆件约束扭转及翘曲刚度。     

薄壁曲梁的横向弯曲稳定分析

根据势能驻值原理,从曲梁的变形几何方程出发,采用转换B3样条函数模拟薄壁杆件横截面的纵向位移场,得到含非线性应变的薄壁曲梁的能量方程,采用样条有限杆元法求解薄壁曲梁的横向弯曲稳定问题。方法很好地描述了薄壁曲梁的翘曲位移和剪滞效应,为分析薄壁曲梁弯扭问题提供了一种有效的方法。数值算例表明本方法的前处理简单、收敛速度快,精度高。     

饱和地基上刚性圆板的扭转振动

用解析的方法首次研究了饱和地基上刚性圆板的扭转振动特性。首先运用Hankel变换求解饱和介质动力问题的控制方程,然后按混合边值条件建立了饱和地基上刚性圆板扭转振动的对偶积分方程,并把对偶积分方程化为第二类Fredholm积分方程。文末数值算例给出了动力柔度系数和扭转角幅值随无量纲频率的变化曲线,并与单相弹性介质情况进行了对比分析。数值结果表明:与经典的弹性介质上基础的振动特性相比,水相的存在对饱和地基上刚性圆板的扭转振动特性有一定的影响,且在共振频率附近可以减弱其振动,当土体渗透性较好时更是如此。     

时域径向积分边界元法在平面单相凝固问题中的应用

该文将时域精细积分边界元方法与界面追踪法相结合,给出平面单相凝固热传导问题的一个有效数值分析方法。首先,利用稳态热传导问题的基本解和径向积分法给出瞬态传热问题的边界积分方程,并采用精细积分方法求解离散的微分方程组,获得相变界面的热流密度。然后应用相变界面上的能量守恒方程,采用界面追踪法来预测相变边界的移动位置,从而给出相关问题数值模拟的结果。最后,为验证该文方法的有效性,给出两个数值算例并与解析解进行了对比。结果表明,该文方法具有较高的求解精度,是求解相变热传导问题的一种有效数值方法。     

考虑剪切变形的薄壁杆件稳定分析

本文提出了一种基于势能原理的薄壁杆件稳定分析的半离散方法。采用转换Ｂ３样条函数作为横截面纵向位移的插值函数，通过变分原理，导出控制微分方程及自然边界条件，利用常微分方程求解器求解。分析时放弃了古典理论关于杆壁中线剪应变为零或剪力流为常数的假设，很好地描述了剪力滞后现象。本方法适用于任意截面形状的薄壁杆件，能够灵活、精确、有效地进行薄壁杆件在轴压与纯弯作用下的稳定分析。算例的快速收敛说明了计算结果的可靠性。     

含曲线裂纹柱体扭转问题的新边界元法

研究了带曲线裂纹柱体的扭转断裂问题,推导出了可以直接应用于任意形状截面含有任意形状曲线裂纹的柱体扭转问题的新的边界积分方程,并建立了带裂纹柱体扭转问题的边界元数值计算方法,提出了裂纹尖端的奇异元和线性元插值模型,给出了抗扭刚度和应力强度因子的边界元计算公式。该文对含有圆弧裂纹、曲线裂纹及直线裂纹的不同截面形状柱体的典型问题进行了数值计算,所得结果证明了边界元方法的正确性和有效性。     

Analyzing interaction between coplanar square cracks using an efficient boundary element‐free method

This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

OPTIMUM SHAPE DESIGN AND POSITIONING OF FEATURES USING THE BOUNDARY INTEGRAL EQUATION METHOD

A design optimization procedure is developed using the boundary integral equation (BIE) method for linear elastostatic two-dimensional domains. Optimal shape design problems are treated where design variables are geometric parameters such as the positions and sizing dimensions of entire features on a component or structure. A fully analytical approach is adopted for the design sensitivity analysis where the BIE is implicitly differentiated. The ability to evaluate response sensitivity derivatives with respect to design variables such as feature positions is achieved through the definition of appropriate design velocity fields for these variables. How the advantages of the BIE method are amplified when extended to sensitivity analysis for this category of shape design problems is also highlighted. A mathematical programming approach with the penalty function method is used for solving the overall optimization problem. The procedure is applied to three example problems to demonstrate the optimum positioning of holes and optimization of radial dimensions of circular arcs on structures.     

The computation of the J integral with the traction boundary integral equation

The solutions of the displacement boundary integral equation (BIE) are not uniquely determined in certain types of boundary conditions. Traction boundary integral equations that have unique solutions in these traction and mixed boundary cases are established. For two‐dimensional linear elasticity problems, the divergence‐free property of the traction boundary integral equation is established. By applying Stokes' theorem, unknown tractions or displacements can be reduced to computation of traction integral potential functions at the boundary points. The same is true of the J integral: it is divergence‐free and the evaluation of the J integral can be inverted into the computation of the J integral potential functions at the boundary points of the cracked body. The J integral can be expressed as the linear combination of the tractions and displacements from the traction BIE on the boundary of the cracked body. Numerical integrals are not needed at all. Selected examples are presented to demonstrate the validity of the traction boundary integral and J integral. Copyright © 2004 John Wiley & Sons, Ltd.     

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.     

An ancillary boundary integral equation for magnetostatic analysis

The boundary element method (BEM) has been established as an effective means for magnetostatic analysis. Direct BEM formulations for the magnetic vector potential have been developed over the past 20 years. There is a less well-known direct boundary integral equation (BIE) for the magnetic flux density which can be derived by taking the curl of the BIE for the magnetic vector potential and applying properties of the scalar triple product. On first inspection, the ancillary boundary integral equation for the magnetic flux density appears to be homogeneous, but it can be shown that the equation is well-posed and non-homogeneous using appropriate boundary conditions. In the current research, the use of the ancillary boundary integral equation for the magnetic flux density is investigated as a stand-alone equation and in tandem with the direct formulation for the magnetic vector potential.     

On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity

The solution of a Dirichlet boundary value problem of plane isotropic elasticity by the boundary integral equation (BIE) of the first kind obtained from the Somigliana identity is considered. The logarithmic function appearing in the integral kernel leads to the possibility of this operator being non-invertible, the solution of the BIE either being non-unique or not existing. Such a situation occurs if the size of the boundary coincides with the so-called critical (or degenerate) scale for a certain form of the fundamental solution used. Techniques for the evaluation of these critical scales and for the removal of the non-uniqueness appearing in the problems with critical scales solved by the BIE of the first kind are proposed and analysed, and some recommendations for BEM code programmers based on the analysis presented are given.     

On the conventional boundary integral equation formulation for piezoelectric solids with defects or of thin shapes

In this paper, the conventional boundary integral equation (BIE) formulation for piezoelectric solids is revisited and the related issues are examined. The key relations employed in deriving the piezoelectric BIE, such as the generalized Green's identity (reciprocal work theorem) and integral identities for the piezoelectric fundamental solution, are established rigorously. A weakly singular form of the piezoelectric BIE is derived for the first time using the identities for the fundamental solution, which eliminates the calculation of any singular integrals in the piezoelectric boundary element method (BEM). The crucial question of whether or not the piezoelectric BIE will degenerate when applied to crack and thin shell-like problems is addressed. It is shown analytically that the conventional BIE for piezoelectricity does degenerate for crack problems, but does not degenerate for thin piezoelectric shells. The latter has significant implications in applications of the piezoelectric BIE to the analysis of thin piezoelectric films used widely as sensors and actuators. Numerical tests to show the degeneracy of the piezoelectric BIE for crack problems are presented and one remedy to this degeneracy by using the multi-domain BEM is also demonstrated.     

Boundary element‐free method (BEFM) for two‐dimensional elastodynamic analysis using Laplace transform

In this paper, we present a direct meshless method of boundary integral equation (BIE), known as the boundary element‐free method (BEFM), for two‐dimensional (2D) elastodynamic problems that combines the BIE method for 2D elastodynamics in the Laplace‐transformed domain and the improved moving least‐squares (IMLS) approximation. The formulae for the BEFM for 2D elastodynamic problems are given, and the numerical procedures are also shown. The BEFM is a direct numerical method, in which the basic unknown quantities are the real solutions of the nodal variables, and the boundary conditions can be implemented directly and easily that leads to a greater computational precision. For the purpose of demonstration, some selected numerical examples are solved using the BEFM. Copyright © 2005 John Wiley & Sons, Ltd.     

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.