Stress–Strain Curve as Extremal of Some Functional

A functional is proposed, the requirement of stationarity of which leads to a differential equation of the second or fourth order with respect to stress as a function of deformation. On the interval of linear and nonlinear elasticity, the spectrum of boundary value problems corresponding to these differential equations is formulated. It is shown that in the nonlinear strain interval, the eigenfunctions of the differential equation are polynomial and power-law similar to the classical Ramberg–Osgood law, which postulates a power-law dependence of strain on stress. For simplicity, this article considers only reversible deformation processes for which the variation formulation reduces to the requirement of stationarity of some functional. Nevertheless, even this simplified formulation allowed a deeper understanding of the mechanisms of deformation beyond the limit of proportionality. In particular, the fact of the successive inclusion of various deformation mechanisms, each of which is determined by a new term in the functional corresponding to this material, can be considered theoretically established. The appearance of a new term in the functional is directly related to the characteristic point on the deformation curve, at which a new mechanism begins to act. As a result, the variation formulation allows one to obtain a theoretical strain curve. This curve satisfies the stationarity requirement of some functional. The theoretical curve was compared with the experimentally obtained tensile curves of materials: steel St3sp (A568M Grade 1023 USA/ASTM), steel 35 (A 682 Grade 1035 USA/ASTM), steel 20 HGR (Grade 900M USA/ASTM), titanium alloy VT6 (Ti-6Al-4V USA/AMS) and aluminum alloy D16 (AA2024 USA/ANSI H35.2) and showed good agreement with the experiment. Thus, a variation model is constructed that allows one to construct the strain curves of various physically nonlinear materials.