Engineering Transactions, 59, 4, pp. 283–297, 2011

YIELD CRITERION ACCOUNTING FOR THE INFLUENCE OF THE THIRD INVARIANT OF STRESS TENSOR DEVIATOR. PART II. ANALYSIS OF CONVEXITY CONDITION OF THE YIELD SURFACE

P. SZEPTYŃSKI
AGH University of Science and Technology Faculty of Mechanical Engineering and Robotics Department of Strength, Fatigue of Materials and Structures

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.
Keywords: yield surface, convexity condition
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