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This paper analyzes the peculiarities of plastic flow of metals for the case of non-proportional loading when the loading path consists of two portions—uniaxial tension and subsequent infinitesimal pure shear (torsion). The issue is discussed from the point of view of the hardening rules governing the kinetics of loading surface. Three cases are considered, flow plasticity theory with isotropic and kinematic hardening rule, as well as the synthetic theory of plastic deformation. As a result, the synthetic theory leads to the results that correlate with experiments, whereas the former two theories associated with smooth loading surfaces give a principal discrepancy with experimental data.

The overwhelming majority of the theories of plastic deformation of metals address the notion of yield and loading surface to give a geometrical interpretation of the onset and development of plastic strains. This paper will analyze (i) plastic flow theories with smooth loading surfaces and (ii) results obtained in terms of the synthetic theory of irrecoverable deformation for the case of a non-proportional loading. Consider a loading path

consisting of two parts in stress space (

Sveshnikova’s experiments were carried out on thin-walled cylinders loaded in uniaxial tension and the additional loading was obtained by the twisting of the specimens.

The goal of the paper is to show that the synthetic theory is capable of describing the occurrence of the increment of plastic deformation due to the additional loading, whereas the theories with smooth loading surfaces lead to the absence of plastic flow, which is contradictive to the experimental results. Although the problem dates back to the 20th century, it remains unsolved till now. The issue of the occurrence of plastic deformation due to an infinitesimal additional loading, nevertheless, is of high importance. Indeed, as is often the case, structural members working under some stress state are subjected to a small additional loading resulted from sudden overloading or lack of fit. Such a situation is typical, e.g. beams deformed by bending and undergoing small torsions.

The occurrence of the increments of plastic strains due to the additional loading is studied in a geometrical way, by means of the analysis of loading surfaces. In terms of the synthetic theory [

Consider the behavior of material modeled by the flow theories based on the isotropic and kinematic hardening rule [

In isotropic hardening, the yield surface increase in size due to the stress vector

same shape, as a result of plastic straining (

where

According to kinematic hardening rule, the yield surface remains the same shape and size but merely translates in stress space (

where

Equations (1) and (2) give the von-Mises yield criterion when

where _{ }is the Kronecker’s delta The length of vector

Now, consider the infinitesimal additional loading

S_{1}-axis (

Therefore, although the theories discussed above are widely used for the modelling of the plastic strains of metals, they are incapable of catching the phenomenon registered in [

The synthetic theory is based on the Batdorf-Budiansky slip concept [

Similarly to the Batdorf-Budiansky concept, the deformation of material is calculated on its two structural levels: macro- and micro-level. A point of a body is considered as an elementary volume of the body,

The modeling of irrecoverable deformation at a point of a body

where

According to Sanders [

The position of plane in

To establish a hardening rule, which governs the kinetics of loading surface during plastic flow, we extend the provision that a surface can be constructed as an inner envelope of planes to the case of loading as well. In the

course of loading, the vector

Each tangent plane corresponds to an appropriate slip system

As it follows from Equation (4) and the hardening rule, material is considered initially isotropic, but after the development of irrecoverable strain its properties (e.g. hardening) become definitely anisotropic.

The condition that a plane in

where the product

An average measure of irrecoverable strain within one slip system

Macro-deformation is defined by a strain vector,

The upper and lower integration limits in (8) are obtained from the condition

Summarizing, the magnitude of plastic deformation rate depends on the set of planes located on the endpoint of

Let us study if an additional plastic strain increment occurs due to the additional loading

This fact means that the action of the

Therefore, in the framework of the synthetic theory, the phenomena of the occurrence of plastic deformation on the orthogonal portion of additional loading can be modelled, this fact is of great importance since is not the case for the flow theories with smooth loading surface.

The formation of corner point (conical singularity) on the loading surface during plastic straining is of crucial importance for the correct formulation of the theories of plasticity. As it has been shown in this paper, the flow plasticity theories based on hardening rules with smooth loading surfaces lead to non-conformity with the experimental result obtained for the case of non-proportional loadings (they give no increment in plastic strain), e.g. when the loading path is a broken line with orthogonal portions. At the same time, the synthetic theory of plastic deformation shows the occurrence of plastic straining in the additional loading even without calculations; it is immediately seen from the shape of loading surface and the direction of additional loading.

The authors expresses thanks to Prof. K. Rusinko (Budapest University of Technology and Economics, Hungary) for many useful conversations on the topics presented in this article.