Effect of the morphology of pores on the regularities of plastic deformation of porous bodies. II. Evolution of the shape of pores in the process of plastic deformation

Abstract

The use of Beltrami's hypothesis, and that means the selection of the elastic constants of a material as the parameters determining the loading surface, makes it possible to replace the solution of the problem of plastic deformation of a real porous semiproduct by the investigation of some mode, medium with recurrent structure and analogous elastic properties. If the shape of the pores and their disposition within the bulk of the material are known, then it is possible, as was shown in [i], to specify the loading surface, and consequently to find the distribution of the mean plastic strain rates. However, during plastic flow the morphology of the pores changes, and so does correspondingly the loading surface. Describing the regularities of such a change is a much more complex task than finding the parameters of the loading surface for the specified configuration of the pores. It is closely connected with the investigation of the flow of the material in the vicinity of the pore, which in itself is of importance. In cold pressing of porous semiproducts the basic mechanism of reducing the pore volume is plastic inflow of material. The nonlinear nature of flow creates considerable difficulties in describing the process. Since within the assumptions of [1] it suffices to examine a body consisting of nonrecurrently arranged equal cells, we will deal with the regularities of the change of shape of a pore within one cell only. The main aim of our research was to shed light on the interrelation between the strain rates of the cell and of the pore. The strain rates of the cell were determined from the rates U x and Uy specified on the lateral sides of the cell (the subscript in the designation of the rate indicates its direction; positive rates are directed toward the center of the cell). Since plastic flow of material does not depend on the actual rates of the process but only on their ratio, we took Uy equal to unity. The rate U x changed within a wide range of positive and negative values. Let us consider plane strain of a cell, an infinite right-angled parallelepiped with a pore in the form of an elliptical cylinder (Fig. 1). Plastic flow was investigated by the finite element method on an ES 10-35 computer. The program made it possible to solve plane and axisymmetric problems. The material of the carcass of the porous body is assumed to be tough and plastic, incompressible, satisfying the condition of Mises plasticity. In the calculations its properties were described by the relations of the model suggested in [2]. The loading surface used in it changes in case of low porosity into a Mises surface; this makes it possible to study the flow of incompressible material, it being regarded as the limit case. Porosity was assumed to be 1%. For each ratio of rates on the sides of the cell Ux/Uy (it is equal to ex/ey because the side of the section of the cell is equal to unity) we found the ratio of the rates on the surface of the pore Vx/Vy. The dependence of Vx/Vy on ex/ey is complex and nonlinear (Fig. 2). We note that hardly anywhere a similarity of deformation of the cell and of the pore was encountered. A more detailed analysis shows that the distribution of rates on the surface of the pore is largely determined by the configuration of the plastic zone originating in flow. In its turn, the configuration of the plastic zone depends more on the geometry of the cell and less on the actual rates on its surface because the form of the plastic zones does not correspond to separate values but to entire intervals of change of the strain rate of the cell. This makes clear the physical meaning of the found dependences and makes it possible to establish some quantitative regularities of the behavior of the pore at different rates on the surface of the cell. Calculations showed that there are four basic plastic zones qualitatively differing from each other. As the horizontal rate U x (Fig. 1) increases from negative to positive values (from tension to compression), the distribution of the rates of the