双压电材料界面力电耦合场奇异性研究

针对不同压电材料中界面裂纹尖端的扇形区域推导出了包含基本方程、裂纹面D-P边界条件和不同压电材料交界面处的边界条件的弱形式。在该弱形式的基础上,利用特征方程展开方法(eigenfunction expansion technique),可以得到一个分析压电材料裂纹尖端处力电耦合场奇异性的特殊的一维有限元列式。该一维有限元列式只需对扇形区域在角度方向上离散,最后的总体方程为一个二次特征根方程。求解该特征根方程就可以得到压电材料裂纹尖端处力电耦合奇异场的特征解。通过数值算例表明该方法可以准确而高效地计算压电材料裂纹尖端处力电耦合奇异场的特征解,进而用该方法研究了双压电材料界面力电耦合场的奇异性。     

用裂纹单元分析双压电材料界面力电耦合奇异场

为了求解双压电材料在机械荷载和(或)外加电场的作用下,界面裂纹尖端的力电耦合奇异场,提出了一种全数值方法。该全数值方法的实施可以分为两个部分:首先,用一维有限元方法求解不同压电材料界面裂纹尖端力电耦合奇异场特征解;然后,采用杂交有限元列式构造一种所谓的裂纹单元,在该杂交有限元的列式中,假设应力场和电位移场是利用上述一维有限元方法计算得到的特征解推导出来的;利用该单元可以得到全部的力电耦合奇异场的解。通过对单一压电材料中心裂纹尖端力电耦合奇异场的计算,该方法的准确性和高效性得到了验证;进而用该方法研究了双压电材料界面力电耦合场奇异场。     

用切口尖端单元分析各向异性材料中多边形孔奇异性应力场

该文提出了一种基于全数值方法的新型杂交元方法, 用于研究各向异性复合材料中多边形孔奇异性应力场干涉问题。该方法的建立分3 个步骤:首先, 用一维有限元方法求解各向异性材料切口尖端奇异性应力场数值特征解;然后, 采用杂交有限元列式构造一种超级切口尖端单元, 其中, 假设应力场和位移场是利用上述奇异性场数值特征解推导出来的;最后, 将上述超级切口尖端单元与传统4 结点杂交应力元组装, 得到新型杂交元方法。算例中, 将裂纹问题作为考核例, 并进一步考察双菱形孔和双矩形孔的奇异性应力干涉问题。算例表明:当前模型能降低单元数, 且精度好;与传统有限元法和积分方程方法相比, 该模型更具有通用性和高效性, 为各向异性材料的细观力学分析打下了基础。     

纤维增强复合材料轴对称界面端的奇异应力场

提出了一种分析横观各向同性纤维增强复合材料轴对称界面端的奇异应力场的特征值法。基于横观各向同性弹性材料空间轴对称问题的基本方程和一阶近似假设,利用分离变量形式的位移函数和无网格算法,导出了关于应力奇异性指数和应力角函数的奇异性特征方程。对于纤维/基体轴对称界面端模型,特征值法给出的应力奇异性指数、相应的位移和应力角函数,与通过有限元分析得到的结果非常吻合。利用有限元计算得到的奇异应力场,结合特征值法给出的应力奇异性指数和应力角函数,通过线性外插得到了相应的应力强度系数。特征值法结合有限元分析,可以完全确定横观各向同性纤维增强复合材料轴对称界面端的奇异应力场。     

铆接界面端三维奇异性应力场的研究

针对铆接结构的特点,应用特征函数扩展技术分析柱坐标下接触界面端的应力奇异性问题。建立了柱坐标下圆柱体端面接触边缘附近的三维渐近位移场和应力场渐近表达式,并根据铆钉/被铆接件接触界面端的位移和应力边界条件,建立一个非线性特征方程组。据此方程组可求解界面端邻域的应力奇异性指数、位移和应力角分布函数的数值解。通过与有限元方法计算结果相对比,验证了该方法的有效性。分析了平头、沉头以及半圆头铆钉构成的铆接结构的应力奇异性问题,考察了铆钉材料、几何形式和摩擦系数对接触界面端应力奇异性指数和应力场角分布的影响。     

反平面载荷作用下带界面裂纹的不同正交异性层板的应力分析

本文对含有边缘界面裂纹的不同正交各向异性平板在反平面载荷作用下的位移场与应力场进行了分析,得到了满足所有基本方程以及裂纹面边界条件与交界面连续条件的位移场与应力场展开式,本文进一步应用变分原理决定应力场展开式中奇异项系数—应力强度因子,计算结果表明,应力强度因子的收敛性是令人非常满意的.      

对称斜交铺层复合材料层板在平面变形情况下分层问题的解析-广义变解法

本文采用弹性力学的位移解法研究对称斜交铺层复合材升层板在平面变形情况下的分层问题,得到了满足所有基本方程,层间连续条件与裂纹表面静力边界条件的位移场与应力场的本征展开式.然后利用分区广义变分原理代替裂纹表面以外的边界条件,确定位移场与应力场表达式中的待定系数,进而确定裂纹尖端附近奇异应力场的控制量--广义应力强度因子.由于所有基本方程预先得以满足,在变分方程中只有线积分而无面积分.计算表明,本文方法前期准备工作简便,计算节省机时,结果收敛迅速.     

对称斜交铺层复合材料层板在平面变形情况下分层问题的解析—广义变解法

本文采用弹性力学的位移解法研究对称斜交铺层复合材升层板在平面变形情况下的分层问题,得到了满足所有基本方程,层间连续条件与裂纹表面静力边界条件的位移场与应力场的本征展开式。然后利用分区广义变分原理代替裂纹表面以外的边界条件,确定位移场与应力场表达式中的待定系数,进而确定裂纹尖端附近奇异应力场的控制量——广义应力强度因子。由于所有基本方程预先得以满足,在变分方程中只有线积分而无面积分。计算表明,本文方法前期准备工作简便,计算节省机时,结果收敛迅速。     

无限弹性基底上热电薄膜的屈曲行为研究

本文研究了热电薄膜粘合到弹性基底结构的屈曲行为．将界面剪切应力和薄膜的轴向应力结合起来,建立了热电薄膜的计算模型,利用边界条件将所求问题转化为一个奇异积分方程．通过使用切比雪夫多项式展开求解奇异积分方程,得到归一化应力强度因子．确定了膜厚度和基材与膜刚度比对薄膜应力和界面应力强度因子的影响．讨论了薄膜长度和厚度比对薄膜应力和界面应力强度因子的影响．结果显示薄膜和基底之间的刚度比对薄膜的应力水平有着较明显的影响．     

垂直界面裂纹应力强度因子的加料有限元分析

为求解裂尖位于界面上的垂直双材料界面裂纹应力强度因子,发展了一种加料有限元方法。该方法应用Williams本征函数展开和线性变换方法求解裂尖渐进位移场,将该位移场加入常规单元位移模式中,得到加料垂直界面裂纹单元和过渡单元的位移模式,给出加料有限元方程。建立了典型垂直界面裂纹平面问题的加料有限元模型,求解加料有限元方程直接得到应力强度因子,与文献结果对比表明该方法具有较高的精度,可方便地推广应用于垂直界面裂纹的计算分析。     

On finite element modelling of surface tension Variational formulation and applications – Part I: Quasistatic problems

This paper describes variational formulation and finite element discretization of surface tension. The finite element formulation is cast in the Lagrangian framework, which describes explicitly the interface evolution. In this context surface tension formulation emerges naturally through the weak form of the Laplace–Young equation.The constitutive equations describing the behaviour of Newtonian fluids are approximated over a finite time step, leaving the governing equations for the free surface flow function of geometry change rather than velocities. These nonlinear equations are then solved by using Newton-Raphson procedure.Numerical examples have been executed and verified against the solution of the ordinary differential equation resulting from a parameterization of the Laplace-Young equation for equilibrium shapes of drops and liquid bridges under the influence of gravity and for various contact angle boundary conditions.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.     

Free vibration analysis of sandwich beams using improved dynamic stiffness method

In this paper, free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness and finite element methods. To determine the governing equations of motion by the present theory, the core density has been taken into consideration. The governing partial differential equations of motion for one element contained three layers are derived using Hamilton’s principle. This formulation leads to two partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are combined to form one ordinary differential equation. Closed form analytical solution for this equation is determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. They are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the known Wittrick–Williams algorithm. After validation of the present model, the effect of various parameters such as density, thickness and shear modulus of the core for various boundary conditions on the first natural frequency is studied.     

Rotation‐free triangular shell element using node‐based smoothed finite element method

In this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second‐order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node‐based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

Predicted axial temperature gradient in a viscoplastic uniaxial bar due to thermomechanical coupling

The thermomechanical response of a uniaxial bar with thermoviscoplastic constitution is predicted herein using the finite element method. After a brief review of the governing field equations, variational principles are constructed for the one-dimensional conservation of momentum and energy equations. These equations are coupled in that the temperature field affects the displacements and vice versa. Due to the differing physical nature of the temperature and displacements, first-order and second-order elements are utilized for these variables, respectively. The resulting semi-discretized equations are then discretized in time using finite differencing. This is accomplished by Euler's method, which is utilized due to the stiff nature of the constitutive equations. The model is utilized in conjunction with stress-strain relations developed by Bodner and Partom to predict the axial temperature field in a bar subjected to cyclic mechanical end displacements and temperature boundary conditions. It is found that spacial and time variation of the temperature field is significantly affected by the boundary conditions. The nomenclature used is given in an Appendix.     

中厚椭球壳自由振动动力刚度法分析

介绍精确动力刚度法分析中厚椭球壳自由振动具体实施方法,据环向波数不同将中厚椭球壳自由振动分解为一系列确定环向波数的一维振动;利用控制方程Hamilton形式建立动力刚度关系,用常微分方程求解器COLSYS求解控制方程获得单元动力刚度,用Wittrick-Williams算法求得该环向波数下椭球壳自振频率。数值算例给出中厚圆球壳及椭球壳不同边界条件的自振频率,验证动力刚度法高效、可靠、精确。     

Elastodynamic problems by meshless local integral method: Analytical formulation

In this paper, analytical forms of integrals in the meshless local integral equation method in the Laplace space are derived and implemented for elastodynamic problems. The meshless approximation based on the radial basis function (RBF) is employed for implementation of displacements. A weak form of governing equations with a unit test function is transformed into local integral equations. A completed set of the local boundary integrals are obtained in closed form. As the closed form of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically. Several examples including dynamic fracture mechanics problems are presented to demonstrate the accuracy of the proposed method in comparison with analytical solutions and the boundary element method.     

Elasto visco-plastic flow with special attention to boundary conditions

In this paper we analyse a simple but non-trivial steady-state creeping elasto visco-plastic (Maxwell fluid) radial flow problem, paying particular attention to the effects of the boundary conditions. Solutions are obtained by integration of a governing equation on stress, using the Runge-Kutta method for initial value problems and finite differences for boundary value problems. A more general approach by the finite element method, which solves for the velocity field rather than the stress field, and which is applicable to a wide range of problems, is presented and tested using the radial flow example.