Nonsteady planar ideal plastic flow: general and special analytical solutions

Abstract

Ideal plastic flows are those for which all material elements follow minimum work paths. For planar flows this implies that two orthogonal families of material lines, called principal lines, are perpetually tangent to the principal strain rate vectors. The general equations for ideal flows in Tresca solids have been given elsewhere as have specific examples of steady plane strain and axisymmetric flows and nonsteady membrane flows. Here we focus on the case of nonsteady plane strain and show that the representation of such flows can be reduced to the solution of a telegraph equation in two independent spatiotemporal characteristic variables. In obtaining this general result, two special cases are lost: those corresponding to the cases where one and two families of characteristics are straight in both Cartesian and principal line spaces. Results for these cases are given as well. Finally, an application to forming process design problems is discussed and a simple example is illustrated using the general solution.