Possible Representation of the Yield Surface and Its Possible Shapes in a Two-Dimensional Problem

Objective: the identification of the shape of the yield surface and analysis of the change in its shape depending on the loading path. The object of research is the basic concept of the theory of plasticity – the loading surface of the material. The subject is the possible representation and shape of the loading surface in a two-dimensional problem. Methods: the article considers a simplified two-dimensional model representing a fragment of a thin-walled pipe. The pipe material is elastic-perfectly plastic material, in accordance with Prandtl diagram. Displacement-Controlled loading for model is assumed. The load is represented by rotation of the pipe cross-section, realized about either a single Cartesian axis or two axes simultaneously. It is shown that load can be considered as a vector. But stress and strain in a point of the body given in term of Tensor Analysis. This issue can be resolved using the Ilyushin representation, according to which a vector is assigned to the deviatoric stress tensor. Results: even for a low level of plastic deformation intensity, the shape and size of the loading surface strongly depend on the loading path. For an isotropic body, this fact can only be explained by the transformation of residual stresses in the body. Practical significance: the proposed approach allows us to apply a simplified mechanical model to the analysis of the transformation of yield surfaces for a wide range of load paths and compare the results with those obtained for real materials.