Convex and Non-convex Flow Surfaces

The strength hypotheses and yield surfaces studied in the classical courses of strength of materials (the normal stress hypothesis, the modes due to von Mises and Tresca) are not sufficient in order to obtain a realistic description of the material behavior at various loading conditions. Therefore advanced models with one or more parameters are suggested in the literature. These models are connected with some limitations of the geometry of the suitable shapes of the surfaces in the stress space. In this paper a compression-torsion diagram normalized with respect to the tension stress is presented. It is possible to compare the models for incompressible material behavior using this diagram. The region of convex forms is bounded by zwei extremal models. In the diagram the meaning of the symmetry in the pi-plane is clarified, which simplifies the comparison of the models. Generalized models (the Radcig-model and convex combination of the models by Sayir and Haythornthwaite) are used to discuss the minimal number of parameters required to obtain convex forms for description of the incompressible material behaviour. The complexity of these models leads to assumption, that they are not suitable for real-world applications. A so-called geometrical-mechanical model with two parameters is proposed, which encompasses a large number of convex geometries. By variation of the power of the stress one tries to obtain the convexity region in the compression-torsion diagram, which is as large as possible. In order to achieve the largest-possible convexity region, the computation of its boundaries is required. Using an equation proposed by Betten-Troost the convexity problem can be investigated analytically. However, some boundaries of the convexity region could not be obtained with this equation analytically. The equation of Betten-Troost is checked by the authors and additional terms are obtained. The restrictions imposed upon the equation are not known, hence the corrected version of the equation is applied. For compressible generalizations the first invariant of the stress tensor is introduced into the models. In dependence of the power of the equivalent stress in the models three simple transformations are proposed. The transformations avoid convexity check in the meridian planes.