© 1952 by Oxford University Press
TWO-DIMENSIONAL PLASTIC STRESS SYSTEMS WITH ISOMETRIC PRINCIPAL STRESS TRAJECTORIES
(
U.S. Naval Research Laboratory Washington, D.C.
U.S. Naval Ordnance Laboratory White Oak, Silver Spring, Md
)
Carathéodory and Schmidt, in their paper on the maximum shear trajectories in plane plastic stress with the yield condition 
(
1–2)/2 = constant, discussed the possibility of these trajectories forming an isometric net. The authors generalize the above study to an arbitrary yield condition f(1, 2) = 0. A complete discussion shows that only for three distinct families of yield conditions, depending on from two to four parameters, do there exist isometric nets satisfying the conditions of the problem beyond the trivial nets consisting of concentric circles and concurrent straight lines. Among these yield conditions the von Mises parabolic yield condition is found to be included. The linear yield condition is discussed in detail; it is shown that = ±+constant and = constant are degenerate cases, the latter falling under elastostatics. The complex functions Z(z) characterizing the possible isometric trajectory patterns are given either explicitly or by equations involving quadratures. It is shown that among the isometric nets which are possible for certain yield conditions are all those which go into themselves either under a rotation or under a translation.