Three-Dimensional Slip-Line Field Theory With Rotational Continuity
For cold-worked metal bodies undergoing plastic deformation along well-defined loading paths, a three-dimensional slip-line field theory is developed, taking into account the new principle of rotation-rate continuity. It is shown that the slip-line network is always confined to one of the three families of principal stress surfaces and that the strain rate normal to those surfaces vanishes. Further, the ratio of the radii of curvature of the slip-lines in the plane tangent to the net remains constant. This condition, in tum, imposes restrictions on the geometric configurations that are allowed for the net boundary. The velocity hodograph always has one of these configurations.
References
R.L. BISH, Plane-stress deformation of a fiat plate by slip between elastic elements, Arch. Mech., 46, 3-12, 1994.
L. PRANDTL, Über die Harte plastischer Körper, Nachrichten von der Königlichen Gesellschaft der Wissenschafen zu Götingen, Mathematisch-Physikalische Klasse, 13, 74-85, 1920.
R. HILL, The mathematical theory of plasticity, Oxford University Press, 1950.
A. NADAI, Plasticity, McGraw-Hill, 1931.
W. PRAGER and P.G. HODGE, Theory of perfectly plastic solids, Chapman and Hall Ltd., 1951.
W. JOHNSON and P.B. MELLOR, Plasticity for mechanical engineers, D. va.n Nostra.nd Co. Ltd., 1962.
W. PRAGER, Recent developments in plasticity, J.Appl. Phys., 20, 235-241, 1949.
W.T. KOITER, Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface, Quart. Appl. Math., 11, 350-354, 1953.
P. PERZYNA, The constitutive equations for rate-sensitive plastic materials, Quart. Appl. Math., 20, 321-332, 1963.
R.L. BISH, Transverse deflection of a cold-worked metal plate clamped around its edge, Arch. Mech., [accepted for publication].
R. VON MISES, Bemerkungen zur Formulierung des mathematischen Problems der Plastizitätstheorie, Zeits. für Angew. Math. und Mech., 5, 147-149, 1925.
H. GEMINGER, Beitrag zum vollständigen ebenen Plastizitätsproblem, Int. Congr. Appl. Mech., 3rd proceedings, Comptes Rendus. Verhandlungen. Stockholm, 2, 185-190, 1930.
W. ROSEHAIN, An introduction to the study of physical metallurgy, Constable and Co. Ltd., London 1915.
L.E. FRENCH and P.F. WEINRICH, The tensile fracture mechanisms of fee metals and alloys - a review of the influence of pressure, J. Aust. Inst. Metals, 22, 40-50, 1977.