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The Problem of Axially Symmetric Flow of Ideal Rigid‐Plastic Huber‐Mises Medium without any Specific Hypotheses
Author(s) -
Rogozińaski M.
Publication year - 1982
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.19820620906
Subject(s) - mathematics , axial symmetry , von mises yield criterion , boundary value problem , mathematical analysis , flow (mathematics) , geometry , finite element method , physics , thermodynamics
In contrast to Shield' or Lippmann' solution in the present paper the Huber‐Hencky‐Mises' yield condition, more suitable for metals than Tresca', is adopted. Moreover, the Haar‐Kármán hypothesis and also any assumptions substituting it, as that used by Lippmann, with the errors introduced by them are avoided. Instead of this, two ratios of stress (or respective strain rate) deviator components to one another are introduced as unknown functions α and β and the parameter λ is eliminated. Then, the well‐known system of seven equations in seven unknown functions for solving the title problem is reduced to a system of two quasilinear partial integro‐differential equations in α and β. The boundary conditions of the problem for both α and β and their first derivatives are found, for a smooth side of the die, on the basis of kinematical and statical conditions. The system of equations is of the mixed type but a transformation of α reduces the problem to the hyperbolic region only. It is possible to find directly the proper solution and not, as usually, its lower and/or upper estimation only.