Determination of the area of mechanical hysteresis loop using mathematical models

Abstract

A method of the hysteresis loop relates to the direct methods for determination of the energy dissipation and studying the inelasticity in the material. The method is based on the direct formation of the mechanical hysteresis loop by static loading and unloading of the sample and measuring of the corresponding deformations. The relative energy dissipation is defined as the ratio of the hysteresis loop area to the elastic energy corresponding to the maximum amplitude of strain. Construction of the hysteresis loop is performed on the installation «torsional pendulum for determination of the mechanical properties of materials» which can work as a device for measuring internal energy dissipation by damped oscillations, and as a precision torsion test machine using a deforming device. The aim of this work is to determine the area of the static hysteresis loop through the choice of the mathematical models of loading and unloading curves with subsequent numerical integration using the ordinate values at equidistant points. The analysis of using polynomials of the second or third degree was carried out according to the criterion of the smallest sum of squared deviations between the empirical and calculated values of the function. The experimentally obtained coordinates of the points of the deformation diagram of the sample during loading and unloading were used as initial data for estimation of regression coefficients in polynomial equations. A distinctive feature of the proposed method is that analytical dependences between stresses and strains obtained by N. N. Davidenkov and containing hard-to-determine geometric parameters of the loop, which must be pre-set from the known values of the logarithmic decrement of oscillations obtained from the experiment are not used in the developed method to calculate the area of the static hysteresis loop. It is shown that a comparative assessment of the relative energy scattering in the ferrite gray iron performed by the direct method of determining the area of the mechanical hysteresis loop at different amplitudes of shear deformation, is in good agreement with the data obtained by the indirect method of damped oscillations on an installation of the similar class.