The nonlinear stress-strain curve model as a solution of the fourth order differential equation

Abstract

The idea that the stress-strain curve satisfies different ordinary differential equations in its different segment is proposed and implemented. In contrast to the classical Ramberg-Osgood law, the model establishes the dependence of stress on deformation. For each segment of the deformation curve, a differential equation is formulated that corresponds to the physics of deformation in this section. The curve is divided into two segment: linear and nonlinear. In the first segment, a second-order differential equation of the simplest type is postulated. In the second segment, a fourth-order differential equation is postulated. The boundary value problem in each segment is formulated from the requirement of continuity and differentiability of the deformation curve. The resulting solution has the advantage that, in contrast to the empirical Ramberg-Osgood law, it has a strictly linear part of the stress-strain curve. The proposed model was tested on modeling the elastoplastic properties of four Russian steels. The standard deviation of the theoretical curves from the samples of the experimental points did not exceed 2.5% for all materials.