Engineering Transactions,
59, 4, pp. 283–297, 2011
10.24423/engtrans.139.2011
P. SZEPTYŃSKI
AGH University of Science and Technology,
Faculty of Mechanical Engineering and Robotics
Department of Strength, Fatigue of Materials and Structures, Kraków
Poland
General form of yield condition for isotropic and homogeneous bodies is considered in the paper. In the space of principal stresses, the limit condition is graphically represented by a proper regular surface which is assumed here to be at least of C2 class. Due to Drucker’s Postulate, the yield surface should be convex. General form of convexity condition of the considered surface is derived using methods of differential geometry. Parametrization of the yield surface is given, the first and the second derivatives of the position vector with respect to the chosen parameters are calculated, what enables determination of the tangent and unit normal vectors at given point, and also determination of the first and the second fundamental form of the considered surface. Finally the Gaussian and mean curvatures, which are given by the coefficients of the first and the second fundamental form as the invariants of the shape operator, are found. Convexity condition of the considered surface expressed in general in terms of the mean and Gaussian curvatures, is formulated for any form of functions determining the character of the surface.
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).
References
D. C. Drucker, A more fundamental approach to plastic stress–strain relations, Proceedings of the first US congress of applied mechanics, American Society of Mechanical Engineers, 487–491, 1952.
A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.
M. Nowak, J. Ostrowska–Maciejewska, R. B. Pęcherski, P. Szeptyński, Yield criterion accounting for the third invariant of stress tensor deviator. Part I. Derivation of the yield condition basing on the concepts of energy-based hypotheses of Rychlewski and Burzyński, Engng. Trans., 59, 4, 273–281, 2011.
B. Raniecki, Z. Mróz, Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect, Acta Mech., 195, 81–102, 2008.