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Rotation‐rate continuity in bi‐axial plastic deformation
Author(s) -
Bish R.L.
Publication year - 2004
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200310098
Subject(s) - slip (aerodynamics) , slip line field , geometry , isotropy , rotation (mathematics) , mathematics , logarithmic spiral , principal axis theorem , plane stress , plasticity , strain rate , mechanics , mathematical analysis , physics , materials science , optics , composite material , finite element method , shear (geology) , thermodynamics
The hypothesis of rotation‐rate continuity, which is associated with the deformation of homogeneous, isotropic bodies by slip, is applied to the plastic flow of a cold‐worked metal (or strain‐aged mild‐steel) under the assumption of bi‐axial strain. In a special case, two classes of rotationally continuous slip‐line fields are shown to exist on curved principal surfaces. The first of these reduces to Hill's equiangular net under plane strain, further special cases of this field being the circular‐centred‐fan and the rectangual‐net. A second class of special solutions consists, again under plane‐strain, of two orthogonally intersecting families of logarithmic spirals. A slip‐line field that occurs in the drawing of a circular cup obeys this condition, as it applies to a curved principal surface. The general condition for continuity of rotation‐rate within a slip‐line net, located on a curved principal surface, is then derived and it is shown that Prandtl's solution to the plane‐strain compression problem obeys this general condition; except in a pair of thin layers adjacent to the platens.