Forming analysis of porous materials

Abstract

A procedure for estimating the applied traction required for forming processes of porous material is suggested. It carries the general structure of the upper-bound approach in the sense that static equilibrium is not enforced and a detailed mechanism of pore closure is not needed. A velocity field for compressible, porosity dependent, plastic continua, is solved for plane and axisymmetric converging flow problems. By employing, in conjunction with the velocity field, Gurson's model (to characterize the effects of the mean stress and the porosity on the yielding of the material), account is taken of the work expenditure due to the dilatancy of the material, beside the other customary dissipation rate terms. The tractions are then evaluated by the limit analysis scheme, leaving the porosity at the exit as a free parameter to be fixed by minimization of the plastic dissipation. It has been found that the dissipation terms depend on the mean normal stresses and the porosity in a highly nonlinear way. In cases where the porosity and the mean stress vary moderately in the domain of the flow, closed-form expressions for the various dissipation rates are provided. They exhibit, in a systematic fashion, the exact features of the corresponding solutions of non-porous materials, but are multiplied by a set of “weight functions” which depend merely on the averaged values of the porosity and the mean normal stress. Parametric study of the presented solutions for several classical processes (wire drawing, extrusion, ironing and rolling) indicates the role of the average porosity in decreasing or increasing the loads (or moments) needed to operate the forming processes, compared to materials with zero porosity. Comparisons to various experiments, numerical solutions (FEM) and to few exact (asymptotic) solutions, validate the feasibility of the suggested scheme for extending the upper-bound analysis to metal forming of porous materials. The fact that forming loads of non-porous material are not necessarily upper bounds to forming loads of porous materials (under the same geometrical and influx speeds) may have valuable consequences.