In most cases the deformation curves that determine the properties of elastoplastic materials are empirical dependencies that interpolate a certain sample of experimental points. At the same time, empirical curves do not provide an idea of the physics of deformation processes in such materials. As an alternative, the article proposes to construct deformation curves as a solution to a differential equation, each term of which has a physical meaning and determines a certain deformation mechanism specific to each material. The offered differential equation is a fourth-order equation with variable coefficients, whose eigenfunctions are polynomials and power functions, which ensures the continuity of this model with the empirical Ramberg-Osgood model. It is shown that the operator of this equation is the Euler operator and is the product of two second-order operators. The first operator defines the mechanism of linear deformation, and the second - the mechanism of nonlinear deformation of the material. The parameters of the material for processing experimental data are determined by the methods of applied mathematics. To reduce the amount of calculations, a method of compressing a two-dimensional (in the coordinates of deformation-stress) region is proposed to find the characteristic point of the deformation curve, which is the proportionality limit. The proposed method for determining the proportionality limit is formal, mathematical and devoid of subjectivity, in contrast to the method prescribed by the current GOST. A consistent physically substantiated system of four boundary conditions on the interval of nonlinear elasticity of the deformation curve is formulated. The mathematical model of building of deformation curves of various physically nonlinear materials allows to create mathematical models of the resource of such materials.
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