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

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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.