Design of Dies of Minimum Length Using the Ideal Flow Theory for Pressure-Dependent Materials

Abstract

This paper develops the ideal plastic flow theory for the stationary planar flow of pressure-dependent materials. Two rigid plastic material models are considered. One of these models is the double-shearing model, and the other is the double slip and rotation model. Both are based on the Mohr–Coulomb yield criterion. It is shown that the general ideal plastic flow theory is only possible for the double slip and rotation model if the intrinsic spin vanishes. The theory applies to calculating the shape of optimal extrusion and drawing dies of minimum length. The latter condition requires a singular characteristic field. The solution is facilitated using the extended R–S method, commonly employed in the classical plasticity of pressure-independent materials. In particular, Riemann’s method is used in a region where all characteristics are curved. It is advantageous since determining the optimal shape does not require the characteristic field inside the region. The solution is semi-analytical. A numerical procedure is only required to evaluate ordinary integrals. It is shown that the optimal shape depends on the angle of internal friction involved in the yield criterion.

General Planar Ideal Flow Solutions with No Symmetry Axis

“…In this case, Equation ( 7) allows for p to be determined as ln p sin ϕ + cos ϕ p 0 sin ϕ + cos ϕ = 2(β − α) sin ϕ, (10) where p 0 is constant. It has been shown in [23] that the plastic flow rule is compatible with the ideal flow condition and the stress solution above if…”

“…where p 0 is constant. It has been shown in [23] that the plastic flow rule is compatible with the ideal flow condition and the stress solution above if It has been shown in [29] that it is always possible to put…”

“…This condition and the equations above constitute an overdetermined system of equations. However, it has been shown in [23] that this system is compatible.…”

“…The present paper adopts the double shearing and rotation model proposed in [22]. Considering steady planar flows, it has been recently proven in [23] that the ideal flow condition is compatible with this material model if the intrinsic spin vanishes. Note that this condition is not satisfied for the model proposed in [13].…”

“…The present paper provides a steady planar ideal flow solution through a non-symmetric die. The theory developed in [23] is adopted. The solution for Tresca's yield criterion is obtained as a particular case.…”