Rotationally continuous plastic flow

Abstract

It is proposed that plastic flow occurs, under certain conditions, in such a way that the rotation-rate vector remains spatially continuous, while Tresca's criterion governs yielding. Solutions associated with this proposal fall into two classes, according to whether the stress point lies on a face or an edge of the hexagonal yield cylinder associated with Tresca's yield criterion. In the former case the solution takes the form of a three-dimensional slip-line field, while in the latter case the stress equations must be referred to the lines of principal stress. Only two types of slip-line field are permitted. Under plane-strain the first of these is Hill's equiangular net, while the remaining class of slip-line fields is constructed from two families of orthogonally intersecting logarithmic spirals in which the coefficients of angle are related reciprocally. Special, or limiting, cases of these allowed fields are the circular centred fan, the rectangular net and the field consisting of two families of orthogonally intersecting logarithmic spirals. For the case where the stress point lies on an edge of the yield cylinder two new theorems are stated.