The stress-strain curve model in the form of an extremal of a non-integrable linear variation form

Abstract

The article develops an idea that the stress-strain curve for an arbitrary material is the extremum of some functional. However, for irreversible processes, the using of the principle of stationarity of some functional is incorrect, because due to the dissipation of the deformation process, the possible work of internal forces is non-integrable. Therefore, it is proposed to use the generalized variational principle of L. I. Sedov for modeling the stress-strain curve of elastoplastic materials. A concept of sequential inclusion of certain deformation mechanisms on different segment of the stress-strain curve is proposed. According to this concept, each section of the stress-strain curve must correspond either to the stationary value of the corresponding functional, or to the stationary value of the non-integrated form of variations of the corresponding stress derivatives. The combination of naturally obtained spectra of boundary conditions at the ends of each segment leads to a variation-consistent formulation of the system of boundary and contact conditions of solutions of different differential equations on each segment of stress-strain curve. As a result, it is possible to construct a differentiable stress-strain curve over the entire area of the stress-strain curve definition. The resulting solution, in contrast to the Ramberg - Osgood empirical law, has a strictly liner segment. The obtained mathematical model was tested on experimental data of materials for various industrial purposes. The achieved accuracy of the mathematical model is sufficient for engineering applications.