不同材料拼接半平面的周期裂缝问题

带周期裂缝的不同材料拼接平面弹性基本问题,马道玮作过讨论,本文将用复变方法讨论拼接半平面上带周期分布裂缝时的弹性基本问题,把在满足一定边界条件下寻求复应力函数的问题归结为求解某种正则型奇异积分方程,并证明这种方程的解存在且唯一。     

基于拉伸振动精确化理论求解含孔厚板弹性波散射问题

针对采用弹性力学平面问题求解波动/振动时常产生较大误差的问题,基于厚板拉伸振动精确化方程,采用复变函数方法对含孔平板中弹性波散射与动应力集中问题进行了研究。利用正交函数展开的方法将待解的问题归结为对一组无穷代数方程组的求解。给出了含椭圆孔厚板拉压弹性波散射与动应力集中的数值结果。研究结果表明:动应力集中系数与分布取决于入射波数、平板厚度、椭圆偏心率等无量纲化参数。     

基于Kriging代理模型的拉压不同模量平面问题的近似求解

该文建议采用Kriging代理模型数值求解拉压不同模量平面问题。通过本构方程光滑化、有限元法及拉丁超立方采样技术,对拉压不同模量桁架与二维平面问题,给出了基于Kriging模型的近似数值解,以代理基于有限元的数值解,并探讨了样本点数目和问题规模对所建Kriging近似模型求解精度/效率的影响。数值算例表明:所提方法可为求解拉压不同模量平面问题提供精度合理的近似数值解。当问题规模较大且正问题需要多次求解时,该方法有望显著减少计算时间,这对于降低拉压不同模量反问题与优化问题的计算开销十分重要。     

带有垂直于极化方向穿透的直裂纹的压电狭长体平面问题的解析解?

通过构造新的广义保角映射,本文利用广义复变函数方法研究了裂纹面上受平面应力和面内电载荷共同作用的带有垂直于极化方向穿透的有限长直裂纹的压电狭长体的平面电弹性问题,给出了电不可渗透边界条件下裂纹尖端应力和电位移强度因子的解析解。结果表明,对于该问题应力场和电场是耦合的。若不考虑电场的作用,则可得到对应纯弹性材料的解析解。当压电狭长体的高度趋于无限大时,所得解析解可退化为已有结果。最后,通过数值算例说明了材料参数、裂纹长度和机电载荷对场强度因子的影响。     

应力边界元法解平面热弹性问题

本文提出了求解平面热弹性问题的应力边界元法。利用应力法由平面热弹性问题的基本方程出发,简要地叙述了边界积分方程的建立,给出了位移单值条件。这种方法适用于应力边界值问题。作为数值计算例,计算了圆形区域和具有偏心圆孔的圆形区域的热应力,得到了满意的结果。应力边界元法也可应用于平板弯曲问题。     

超弹性材料过盈配合的轴对称平面应力解答

主要讨论了一种采用超弹性材料轴对称过盈配合的橡胶减震元件,推导了橡胶材料过盈配合平面应力的大变形解析解,并应用 ABAQUS 有限元软件进行数值模拟,分析和对比了解析解和数值解的结果,定量分析材料泊松比和体积刚度对问题的影响。     

出平面线源荷载对半空间中半圆形凸起的圆柱形弹性夹杂的散射

采用复变函数法和Green函数法,求解出平面线源荷载对半空间中半圆形凸起的圆柱形弹性夹杂的散射。首先,给出在含有半圆形凸起的圆柱形弹性夹杂的弹性半空间中,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函数,取该位移函数作为Green函数;然后,采用分区的思想,分别构造出夹杂内的驻波和夹杂外的散射波,满足"公共边界"处位移和应力的连续性条件,建立起求解该问题的无穷代数方程组;最后,给出了动应力集中系数和水平地面位移幅值的数值结果,并进行了讨论。     

板件中间V形加劲复杂卷边槽钢轴压构件的稳定性能研究

分别采用有限条数值解法和手算方法对16根轴压简支试件的弹性局部、畸变屈曲临界应力进行了计算,对比了数值解法与手算方法的计算结果,验证了手算方法求解弹性稳定临界应力的有效性。研究了V形加劲复杂卷边槽钢轴压构件的弹性稳定性能。研究表明:V形加劲肋的存在对轴压试件的弹性局部屈曲临界应力影响显著,但对弹性畸变屈曲临界应力影响较小,减小翼缘板件的宽厚比能明显提高截面的弹性畸变屈曲临界应力。     

纤维束张紧力缠绕复合材料飞轮初应力的三维数值分析

利用平面应力型全弹性模型的思想(即将纤维束张紧力缠绕看成多层复合材料薄环连续过盈装配的过程) , 建立了三维纤维束张紧力缠绕复合材料飞轮初应力分析模型, 并给出了基于面-面接触算法求解张紧力缠绕复合材料飞轮初应力的三维数值方法。算例分析表明, 三维数值分析得到的飞轮的环向初应力及径向初始压应力(数值) 均略低于平面应力模型的结果, 且这种差距随着飞轮轴向长度的增加而缓慢增大; 三维分析证实了平面应力模型关于张紧力缠绕复合材料飞轮的初应力分析有足够的精度。最后给出了三维模型轴向效应的表征方法。      

复变函数方法分析剪－压组合作用圆孔平面的弹塑性应力集中问题

本文引用复变函数保角变换方法分析剪－压组合作用圆孔平面的弹塑性应力集中问题。本文发展了伽林等用复变函数求解塑性应力的方法，在更复杂的受力条件下给出应力集中函数及弹塑性界限形状。     

The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics

 The meshless local Petrov-Galerkin (MLPG) approach is an effective method for solving boundary value problems, using a local symmetric weak form and shape functions from the moving least squares approximation. In the present paper, the MLPG method for solving problems in elasto-statics is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method, as the essential boundary conditions can not be enforced directly when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present MLPG method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.     

A Meshless Local Petrov-Galerkin Method for the Analysis of Cracks in the Isotropic Functionally Graded Material

A meshless local Petrov-Galerkin method (MLPG) [[Atluri and Zhu (1998)] for the analysis of cracks in isotropic functionally graded materials is presented. The meshless method uses the moving least squares (MLS) to approximate the field unknowns. The shape function has not the Kronecker Delta properties for the trial-function-interpolation, and a direct interpolation method is adopted to impose essential boundary conditions. The MLPG method does not involve any domain and singular integrals to generate the global effective stiffness matrix if body force is ignored; it only involves a regular boundary integral. The material properties are smooth functions of spatial coordinates and two interaction integrals [Rao and Rahman (2003a,b)] are used for the fracture analysis. Two numerical examples including both mode-I and mixed-mode problems are presented to calculated the stress intensity factors (SIFs) by the proposed method. Example problems in functionally graded materials are presented and compared with available reference solutions. A good agreement obtained show that the proposed method possesses no numerical difficulties.     

SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations

We propose a new and simple technique called the Symmetric Smoothed Particle Hydrodynamics (SSPH) method to construct basis functions for meshless methods that use only locations of particles. These basis functions are found to be similar to those in the Finite Element Method (FEM) except that the basis for the derivatives of a function need not be obtained by differentiating those for the function. Of course, the basis for the derivatives of a function can be obtained by differentiating the basis for the function as in the FEM and meshless methods. These basis functions are used to numerically solve two plane stress/strain elasto-static problems by using either the collocation method or a weak formulation of the problem defined over a subregion of the region occupied by the body; the latter is usually called the Meshless Local Petrov–Galerkin (MLPG) method. For the two boundary-value problems studied, it is found that the weak formulation in which the basis for the first order derivatives of the trial solution are derived directly in the SSPH method (i.e., not obtained by differentiating the basis function for the trial solution) gives the least value of the L²-error norm in the displacements while the collocation method employing the strong formulation of the boundary-value problem has the largest value of the L²-error norm. The numerical solution using the weak formulation requires more CPU time than the solution with the strong formulation of the problem. We have also computed the L²-error norm of displacements by varying the number of particles, the number of Gauss points used to numerically evaluate domain integrals appearing in the weak formulation of the problem, the radius of the compact support of the kernel function used to generate the SSPH basis, the order of complete monomials employed for constructing the SSPH basis, and boundary conditions used at a point on a corner of the rectangular problem domain. It is recommended that for solving two-dimensional elasto-static problems, the MLPG formulation in which derivatives of the trial solution are found without differentiating the SSPH basis function be adopted.     

A meshless local Petrov–Galerkin method for large deformation contact analysis of elastomers

A meshless method based on the local Petrov–Galerkin formulation is applied to the large deformation contact analysis of elastomeric components. Trial functions are constructed using the radial-basis function (RBF) coupled with a polynomial-basis function. The plane stress hypothesis and a pressure projection method are employed to overcome the incompressibility or nearly incompressibility in the plane stress and plane strain problems, respectively. Two different sets of equations are used for the nodes on the contact surface and nodes not on the contact surface, respectively, which is based on the meshless local Petrov–Galerkin method (MLPG) establishing equations node by node. Numerical results for several examples show that the present method is effective in dealing with large deformation contact problems.     

An extension of Key's principle to nonlinear elasticity

A variational principle for finite isothermal deformations of anisotropic compressible and nearly incompressible hyperelastic materials is presented. It is equivalent to the nonlinear elastic field (Lagrangian) equations expressed in terms of the displacement field and a scalar function associated with the hydrostatic mean stress. The formulation for incompressible materials is recovered from the compressible one simply as a limit. The principle is particularly useful in the development of finite element analysis of nearly incompressible and of incompressible materials and is general in the sense that it uses a general form of constitutive equation. It can be considered as an extension of Key's principle to nonlinear elasticity. Various numerical implementations are used to illustrate the efficiency of the proposed formulation and to show the convergence behaviour for different types of elements. These numerical tests suggest that the formulation gives results which change smoothly as the material varies from compressible to incompressible.     

Meshless local Petrov–Galerkin (MLPG) method in combination with finite element and boundary element approaches

The meshless local Petrov–Galerkin (MLPG) method is an effective truly meshless method for solving partial differential equations using moving least squares (MLS) interpolants. It is, however, computationally expensive for some problems. A coupled MLPG/finite element (FE) method and a coupled MLPG/boundary element (BE) method are proposed in this paper to improve the solution efficiency. A procedure is developed for the coupled MLPG/FE method and the coupled MLPG/BE method so that the continuity and compatibility are preserved on the interface of the two domains where the MLPG and FE or BE methods are applied. The validity and efficiency of the MLPG/FE and MLPG/BE methods are demonstrated through a number of examples. Received 6 June 2000     

A quadrilateral shell element incorporating thickness-stretch for nearly incompressible hyperelasticity

The authors proposed a quadrilateral shell element enriched with degrees of freedom to represent thickness-stretch. The quadrilateral shell element can be utilized to consider large deformations for nearly incompressible materials, and its performance is demonstrated in small and large deformation analyses of hyperelastic materials in this study. Formulation of the proposed shell element is based on extension of the MITC4 shell element. A displacement variation in the thickness direction is introduced to evaluate the change in thickness. In the proposed approach, the thickness direction is defined using the director vectors at each midsurface node. The thickness-stretch is approximated by the movements of additional nodes, which are placed along the thickness direction from the bottom to the top surface. The transverse normal strain is calculated using these additional nodes without assuming the plane stress condition; hence, a three-dimensional constitutive equation can be employed without any modification. In this work, the authors apply an assumed strain technique to the special shell element to alleviate volumetric locking for nearly incompressible materials. Several numerical examples are presented to examine the effectiveness of the proposed element.     

Meshless local Petrov–Galerkin method-higher Reynolds numbers fluid flow applications

The meshless local Petrov–Galerkin (MLPG) primitive variable based method is extended to analyze the incompressible laminar fluid flow within or over some different two-dimensional geometries. Although still in laminar regions, the Reynolds numbers considered in this study are in the ranges for which, in the literature, the MLPG primitive variable based method has never produced stable solutions and comparable results with those of the conventional methods. The considered test problems include, a steady lid-driven cavity flow with Reynolds numbers up to and including 10,000, a flow over a backward-facing step at 800 Reynolds number, and a transient fluid flow past a circular cylinder with Reynolds numbers up to and including 200. The present method solves the incompressible Navier–Stokes (N–S) equations in terms of the primitive variables using the characteristic-based split (CBS) scheme for discretization. The weighting function in the weak formulation of the governing equations is taken as unity, and the field variables are approximated using the moving least square (MLS) interpolation. For validation purposes, the obtained results are compared with those of the conventional numerical methods. The agreements of the compared results reveal a step forward towards further applications of the MLPG primitive variable based approach.     

A MLPG4 (LBIE) Formulation in Elastostatics

Very recently, Vavourakis, Sellountos and Polyzos (2006) ({CMES: Computer Modeling in Engineering {\&} Sciences, vol. 13, pp. 171--184}) presented a comparison study on the accuracy provided by five different elastostatic Meshless Local Petrov-Galerkin (MLPG) type formulations, which are based on Local Boundary Integral Equation (LBIE) considerations. One of the main conclusions addressed in this paper is that the use of derivatives of the Moving Least Squares (MLS) shape functions decreases the solution accuracy of any MLPG(LBIE) formulation. In the present work a new, free of MLS-derivatives and non-singular MLPG(LBIE) method for solving elastic problems is demonstrated. This is accomplished by treating displacements and stresses as independent variables through the corresponding local integral equations and considering nodal points located only internally and externally and not on the global boundary of the analyzed elastic structure. The MLS approximation scheme for the interpolation of both displacements and stresses is exploited. The essential displacement and traction boundary conditions are easily satisfied via the corresponding displacement and stress local integral equations. Representative numerical examples that demonstrate the achieved accuracy of the proposed MLPG(LBIE) method are provided.     

A VDQ-based multifield approach to the 2D compressible nonlinear elasticity

A numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu-Washizu principle, considering displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix-vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ-based approach. Some well-known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.