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SUMMARYA procedure for solving quasi-static large-strain problems by the material point method is presented. Owing to the Lagrangian-Eulerian features of the method, problems associated with excessive mesh distortions that develop in the Lagrangian formulations of the finite element method are avoided. Threedimensional problems are solved utilizing 15-noded prismatic and 10-noded tetrahedral elements with quadratic interpolation functions as well as an implicit integration scheme. An algorithm for exploiting the numerical integration procedure on the computational mesh is proposed. Several numerical examples are shown.

The problem of ow of a granular material during the process of discharging a silo is considered in the present paper. The mechanical behaviour of the material is described by the use of the model of the elasticplastic solid with the Drucker-Prager yield condition and the non-associative ow rule. The phenomenon of friction between the stored material and the silo walls is taken into account-the Coulomb model of friction is used in the analysis. The problem is analysed by means of the particle-in-cell method-a variant of the ÿnite element method which enables to solve the pertinent equations of motion on an arbitrary computational mesh and trace state variables at points of the body chosen independently of the mesh. The method can be regarded as an arbitrary Lagrangian-Eulerian formulation of the ÿnite element method, and overcomes the main drawback of the updated Lagrangian formulation of FEM related to mesh distortion. The entire process of discharging a silo can be analysed by this approach. The dynamic problem is solved by the use of the explicit time-integration scheme. Several numerical examples are included. The plane strain and axisymmetric problems are solved for silos with at bottoms and conical hoppers. Some results are compared with experimental ones.In the past, the problem was analysed under some assumptions leading to signiÿcant simpliÿcation, or even disregard of the principles of continuum mechanics. Simple expressions for stress and velocity distributions are given, for example by Jenike and Shield [1] and Walters [2].Recently, the problem has been analysed on the basis of continuum mechanics by several authors. The related initial-boundary value problems are solved using numerical techniques, in particular the ÿnite element method. As pioneering works on this ÿeld, those written by Eibl and co-workers [3; 4] should be cited here. In the ÿnite element method, two di erent descriptions of motion have been used in the analysis of the considered problem: the Eulerian description, and the Lagrangian one. In the Eulerian description, the granular material is treated as a non-classical uid [3-6]the analysis is most useful in the case of continuous reÿlling a silo with the material. When the Lagrangian description is used [6][7][8], the solid model of the granular material is applied; in this case the whole discharging process can be analysed.Another approach, based on the discrete element formulation, has been suggested recently by Langston et al. [9;10], where material grains are modelled as spherical and cylindrical particles. This technique seems to be useful when the ratio between the diameters of the silo outlet and the material grain is not high.In the present paper, the entire process of discharging a silo is analysed. The ow of a granular material in a silo is highly distorted. This causes severe problems in computational modelling of such a process. When the updated Lagrangian formulation of the ÿnite element method is used to describe this large displacement, large strain problem, the original...