On the Determination and Use of Principal Lines
Assuming plane strain for Huber-Mises yield condition (for a Coulomb-Tresca one, both plane strain and stress are admitted) of incompressible homogeneous isotropic ideal plastic medium, basic relation of principal lines (of curvatures and their derivatives), consisting with known relations is derived. It leads, in the case of family of curves translated to one another, to an ordinary differential equation of the second order in y1 where y1 (x1) denotes principal line. The solution of this equation is known (the respective integrals are tabulated in the paper) and thus the result – fully consisting with the well-known Prandtl solution is obtained by the direct method. Similarly in the case of family of homothetic principal lines analogical (although more complicated) equation is obtained in t=dq/dj, where q=ln r and r, j are polar coordinates. In this case the principal lines are directly proved to be logarithmic spirals with an arbitrary slope with respect to radii vectors (the latter case is degenerated one) or other spirals - that is presumably a "new" solution -which, for some values of parameters, may be fairly good approximated by Galileo spirals. Calculation and displaying have been performed by means of ODRA 1204 computer. A brief note on possible degenerations is added.
H. LIPPMANN, Principal line theory of axially symmetric plastic deformation, J. Mech. and Phys. of Solids, 10, 2, 111-122, 1962.
H. LIPPMANN, Statistic and dynamics of axially symmetric plastic flow, ibidem, 13, 1, 29/39, 1965.
Z. MARCINIAK, Z. MRÓZ, W. OLSZAK, P. PERZYNA, J. RYCHLEWSKI, A. SAWCZUK i W. URBANOWSKI, Wprowadzenie w teorie plastyczności, Warszawa 1962.
G. DARBOUX, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, t. 1, Paris 1910.
E. KAMKE, Differentialgleichungen, Lösungsmethoden und Lösungen, I. Gewöhnliche Differential-gleichungen, Leipzig 1959.
G. LORIA, Spezielle algebraische und transzendente ebene Kurven, Leipzig und Berlin, 1911.
Copyright © 2014 by Institute of Fundamental Technological Research
Polish Academy of Sciences, Warsaw, Poland