Plane Strain Compression of Rigid-Perfectly Plastic Strip Between Parallel Dies With Slipping Friction

Abstract

This paper presents the results of computer calculations of a class of slipline solutions for compression between parallel dies with slipping friction at the die-metal interface such that the frictional shear traction is a constant proportion of the yield stress. The slipline fields considered here have previously only been suggested qualitatively. The fields are of “indirect type”, requiring the solution of linear integral equations. They have been analyzed and computed here using the recently developed matrix operator procedure. The numerical results obtained are compared with those obtained from approximate upper bound and other “technological” theories.

A numerical method for determining the strain rate intensity factor under plane strain conditions

“…Those are the β-line between the points Strain rate intensity factor under plane strain conditions The dependence of the dimensionless strain rate intensity factor on x is shown in Fig. 12 at several values of w/ h. All the dashed lines correspond to the value of d found from (33). In contrast to this approximate solution ignoring end effects, the strain rate intensity factor for a finite layer varies along the friction surface.…”

“…Then, it follows from (24) and (46) that the strain rate intensity factor is given by (33). Substituting (35) and (42) into (43) and taking into account that S ≤ 0 according to the convention adopted and φ varies in the range (36) yield…”

“…Substituting this value of u β and the derivative ∂u α /∂β from (53) into (15) gives E S = 4V / √ 3. Then, it follows from (19) and (51) that the strain rate intensity factor is given by (33).…”

“…3) is found. The numerical solution for stress and velocity in this process has been obtained in [11] assuming the maximum friction law and in [33] assuming that the frictional shear traction on the plate faces is a constant proportion of the shear yield stress. Therefore, the main contribution of the present solution is the distribution of the strain rate intensity factor along the friction surface.…”

“…9 by the broken line. The solid line corresponds to the exact value given by (33). Introduce the relative error by δ = |d e − d n |/d e where d e is the exact value of d from (33) and d n is the value of d from the numerical solution.…”