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

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.

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

**Full Text:**PDF

Copyright © 2014 by Institute of Fundamental Technological Research

Polish Academy of Sciences, Warsaw, Poland