Novel exact solutions of the Duffing equation: stability analysis and application to real non-linear deformation tests

Abstract

In this study, novel exact solutions of the Duffing equation with their phase portraits have been proposed and reasoned. It is shown that phase trajectories are initially elliptical and become distorted in the unstable area within the growth of the variable parameter. Instability criteria of identified solutions have been determined together with the Fourier series transformation up to the first and high harmonics in a sense of the physical interpretation. An explicit form for the differential operator, corresponding to considered functions, has been derived with evaluation of its main functional spectrum. Non-isothermal creep tests of different materials were completely described using the Duffing equation via noted solutions up to the fracture as processes with personal deformation response. We successfully examined a relationship between the thermal and magnetic properties of the ferromagnetic amorphous alloy under its non-linear deformation, using the critical exponents. With a high linear correlation between our model and experiments, behaviour of organic and metallic systems is well predicted at the same thermo-mechanical testing conditions on the mesoscale.