Development of Flow Element Method (FLEM) for large deformation and flow problems of a continuum

本文介绍了连续体大变形分析的流动单元法(FLEM)的实用公式(Kiyama，H.等，1991)。这种数值方法的基本思想源于离散单元法(DEM)的原理(Cundall，P.A.，1971)。运动方程是节点位移的控制表达式。方程的求解采用DEM的显式时间推进格式。该方案避免了大量的矩阵运算。作者报道了这种方法和数值算例(Kiyama，H.等人，1995)。采用所谓的Jaumann应力率来建立FLEM，其中旋转的表达式被导出为涡量的显式函数。这一应激率已被广泛用作客观应激率之一。然而，许多研究人员在简单剪切模拟中观察到了不切实际的应力响应。人们在这方面做了大量的工作，研究了简单剪切中涡量与变形梯度之间的关系。用有限元方法对平面应变条件下端部位移的二维弹性块体进行了数值模拟，研究了应力率对变形和应力状态的影响，并在此基础上引入了两个坐标系来计算应力率的修正项。一个是运动方程的全局坐标。另一种是一组局部坐标，每个坐标的原点分别附加到单元中的高斯点上，并位于全局系统中。后者用于处理元素的旋转。最后，成功地建立了FLEM的实用公式，用于模拟大剪切变形和流动问题。

This report describes a practical formulation of Flow Element Method (FLEM)(Kiyama, H.et al., 1991) for large deformation analysis of continua. The basic idea of this numerical method has originated from the principle of Distinct Element Method (DEM)(Cundall, P.A., 1971). The equation of motion is the governing expression for displacements of nodes. The explicit time-marching scheme of DEM is adopted to solve the equation. This scheme enables to avoid large matrix computations. This procedure and numerical examples have been reported by the authors (Kiyama, H.et al., 1995).The so-called Jaumann rate of stress is adopted to formulate FLEM, in which the expression for rotation is derived as an explicit function of the vorticity. This stress rate has been widely used as one of objective stress rates. However, many investigators have observed unrealistic stress responses in simple shear simulation. Considerable effort has been expended on this subject.The relation between the vorticity and the deformation gradient in simple shear is studied. A two-dimensional elastic block subjected to end-displacement in a plain strain condition is also simulated by the FLEM.These are numerical studies on effects of the stress rate on deformations and states of stress.On the basis of this consideration, we introduce two coordinate systems to calculate correction terms in the stress rate. One is the global coordinate for the equation of motion. The other is a set of local coordinates, each origin of which is respectively attached to Gaussian points in an element and is located in the global system. The latter is meant to treat rotation of element. Finally, the practical formulation of FLEM is successfully established to simulate large shear deformation and flow problems.