Yield Criterion Accounting for the Influence of the Third Invariant of Stress Tensor Deviator. Part II. Analysis of Convexity Condition of the Yield Surface - SZEPTYŃSKI, P. (Engineering Transactions, 2011)
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.