Abstract

Abstract

This paper presents the solutions for the plane-strain extrusion of porous material. We consider the problem of a stationary plastic flow through a wedge shaped die. We neglect friction between the die and the deformed material since it is rather a negative effect and should be avoided in manufacturing. The elliptic Green type yield condition and its piecewise-linear approximation are adopted for this problem. In the last case, we obtain analytical solution that links extrusion pressure and area reduction to the initial and final density of the porous material. For elliptic Green yield condition the problem reduced to nonlinear ODE that integrated numerically. The results are compared with known solution for Gurson model. The extrusion pressure predicted by the piecewise-linear model is lower than what obtained by the elliptic Green model. In turn, the pressure predicted by elliptic Green model is lower than the pressure obtained in the frame of Gurson model. At low values of area reduction, all three models predict approximately the same extrusion pressure. With a small initial porosity of the material, the Gurson model gives results that are close to the elliptic Green model, and with a large initial porosity, to the piecewise-linear Green model.

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Extrusion is a valuable technological process that has long been used for continuous metal processing as well as in pharmacy and food industry [1-3]. When the die walls are smooth enough and the taper angle is small, a radial flow of the material is realized during extrusion. Plane strain radial plastic flow is one of the classical problems in the theory of plasticity. The first known solution was obtained by Nadai [4], who determined the stress field in an ideal plastic material. The stationary velocity field corresponding to this solution was found by Hill [5] and, independently, by Sokolovsky [6]. Sokolovsky also found a complete solution to the problem for the material with power-law hardening according to the Hollomon equation. The result of Durban and Budiansky [7] is obtained for linear-hardening material (Ludwik equation). Haddow and Danyluk obtained an elastic-plastic solution of the same problem for non-hardening material in the framework of the Prandtl - Reuss theory [8]. Some other analytical and numerical results can be found in [9-12]. All the mentioned results were obtained for plastically incompressible materials. Plastically compressible (porous) materials can be described by the classical Mohr - Coulomb, Drucker - Prager, or Mises - Schleicher yield conditions. Although the associated plasticity for these yield conditions can leads to known discrepancies between the calculated volumetric strain and the experimental one, they often used to construct the models of complex media on the base of micromechanical solutions [13-15]. More precise yield conditions (for example, Green type models [16] and Gurson model [17]) are explicitly depend on the relative density (or porosity) of the material. For the Gurson model, an approximate analytical solution (of the first order in porosity) for stationary plane-strain radial plastic flow is known [18]. The Gurson model is also utilized to analyze plane-strain extrusion in [19-21]. Numerical results for the anisotropic model are presented in [22]. For Green type models, a number of results for axisymmetric extrusion were presented [23-25]; also the analytical solution [26; 27] for equal channel angular extrusion can be mentioned. The present paper provides the solutions to the plane-strain problem of a stationary plastic flow through a wedge shaped die (Fig. 1). For the piecewise-linear Green type criterion, an exact analytical solution is obtained. For elliptic Green yield condition the problem reduced to nonlinear ODE that integrated numerically. As in [18; 28], friction between the die and the deformed material is neglected since it is rather a negative effect and should be avoided in manufacturing. The results are compared with solution [18] for Gurson model. Fig. 1. Plane-strain extrusion through a wedge-shaped die 1. Plane-strain radial flow The problem with cylindrical symmetry is considered. The radial flow is described by velocity vector , . It is assumed that the cylindrical surface is a free boundary (see Fig. 1). The strain rate tensor has the following form For stationary flow, the continuity equation is satisfied, where is the time, is the dimensionless density of the porous material (the value corresponds to a porosity-free material), is the Hamiltonian. Hence, for the problem under consideration, it follows that . (1) From (1) it follows that (2) The equilibrium equation with respect to (2) takes the form . (3) From (2) it also follows that density distribution obeys the equation . (4) 2. Green type elliptic yield criterion We utilize the elliptic Green yield condition and the associated flow rule , . (5) where functions and are shear and volumetric plastic moduli, respectively; is the mean stress, is the shear stress intensity, , is the (macroscopic) Cauchy stress tensor, is a scalar plastic multiplier. From (5) it follows that (see Appendix A) , where is the unit tensor, . Hence (6) where . Substituting (6) into (3), one can obtain nonlinear first-order ODE that determines the function . Boundary condition is according to (6) since the surface is traction-free, i.e. . With calculated , formulas (4) and (6) determine density and pressure in the channel. 3. Green type piecewise-linear yield criterion Under the plane-strain condition, the following piecewise-linear criterion can be utilized: . (7) Here and are the largest and smallest eigenvalues of the stress tensor, respectively. In the problem under consideration it is reasonable to assume that , , and according to (7) the following is obtained . (8) The normality rule associated with (7) leads to the expressions , , . (9) and according to (4) . (10) Taking into account the equalities (8) and (9), the equilibrium equation (3) takes the form (11) and hence . (12) Equations (10) and (12) define the pressure distribution in the channel in a parametric form with the parameter . This solution is valid for , where can be determined from (12) with . When , the stress state at the inlet of a channel is hydrostatic compression. 4. Results and discussion The model [29] was utilized to determine the plastic modules: , . where is the shear yield stress of a porosity-free material. Fig. 2 shows the dimensionless extrusion pressure , where , versus area reduction calculated according the obtained solutions for different values of initial density. Fig. 3 shows the density of the material at the inlet of the channel, required to achieve the specified values of relative density at the outlet of the channel. For comparison, we write down an approximate solution [18] for the Gurson model , , , . Fig. 2. Extrusion pressure and area reduction required to achieve the specified values of final density: x - Gurson model (solution [18]), blue line - elliptic Green type model (numerical solution), red line - piecewise-linear Green type model (eqs. (10) and (12)), dash line corresponds to (incompressible von Mises material) Fig. 3. Initial density and area reduction required to achieve the specified values of final density: x - Gurson model (solution [18]), blue line - elliptic Green type model (numerical solution), red line - piecewise-linear Green type model (eq. (10)) It should be noted regarding the elliptic Green model and Gurson model that extrusion pressure depends nonmonotonically on the area reduction R. With R above a certain value, there is a zone near the channel inlet where the pressure increases in the direction of material motion. This effect is pronounced for the Gurson model and is barely noticeable for the elliptical Green model. In addition, for both models, when area reduction is above a certain value, the compaction begins outside the tapered region. The extrusion pressure predicted by the piecewise-linear model is lower than what obtained by the elliptic Green model. In turn, the pressure predicted by elliptic Green model is lower than what can be calculated by solution [18], obtained in the frame of Gurson model. At low values of area reduction R, all three models predict approximately the same extrusion pressure. With a small initial porosity of the material, the Gurson model gives results that are close to the elliptic Green model, and with a large initial porosity, to the piecewise-linear Green model. Appendix A. Stress derivation for Green model For the yield condition ; , the normality rule leads to the following expression for plastic strain rate tensor [30] (13) Applying the trace operator to both sides of equality (13), we find . (14) and . Substituting the last expression in (13), we express the stress tensor as (15) and then . Applying the trace operator to both sides of the last equality, we find and then . Substituting this expression together with (14) into the yield condition, we obtain the following expression for the plastic multiplier . Substituting this expression into (15), after some algebra, we have , .