On two-level models of the Taylor - Bishop - Hill type for describing the elastoplastic deformation of polycrystalline bodies: one option for solving the problem of uncertainty in the choice of active slip systems

One of the first two-level physically oriented models intended to describe plastic deformation was the rigid-plastic model of J.I. Taylor, the mathematical justification of which was subsequently presented in the works of J. Bishop and R. Hill. Various versions of models based on the main provisions of these pioneering works are usually called in the literature models of the Taylor - Bishop - Hill (TBH) type. Despite the prevalence of TBH-type models, they have disadvantages (the presence of a constraint (connection), which is the condition of incompressibility, the uncertainty of choosing a set of five slip systems when the condition of activation of six or more systems is met). Taking into account elastic deformations, introduced in the later model of T.G. Lin, made it possible to overcome the disadvantage associated with the presence of a constraint. At the same time, it became possible to realize elastic-plastic deformations when less than five slip systems are activated. However, the most important disadvantage is the uncertainty in choosing a set of active slip systems - remains. It should be emphasized that the limitation of the number of slip systems to five when the representing point in the stress space hits a vertex of a higher order than the fifth is due only to the procedure for solving the velocities (or increments) of shears and stresses. There is no physical justification for such a limitation. In this regard, since the 70s of the twentieth century, two-level elastoviscoplastic (i.e., strain rate-dependent) models have become most widespread. It was shown that when the velocity sensitivity parameter tends to zero, the resulting solution converges to the solution of the elastoplastic model. However, in this case, the system of equations of the constitutive model becomes rigid, which leads to the need to use implicit integration schemes and a significant decrease in computational efficiency. Taking this circumstance into account, numerous attempts have been made to get rid of this most important disadvantage of TBH-type models, however, the options known to the authors are reduced to various mathematical procedures that do not have proper physical justification. In this work, we propose a version of a physically based elastic-plastic model that uses the basic provisions of TBH-type models, but is free from the disadvantages noted above. When more than 5 slip systems are simultaneously activated, they are all considered “equal in rights” for the implementation of plastic deformation by shear. An iterative procedure is proposed to determine the rates (increments) of shears for all slip systems that are potentially active at the moment of deformation under consideration.