Novel Exact Solutions of the Duffing Equation: Stability Analysis and Application to Real Non-Linear Deformation Tests

In this study, novel exact solutions of the Duffing equation with their phase portraits are proposed and reasoned. It is shown that phase trajectories are initially elliptical and become distorted in the unstable area with the growth of the variable parameter in the damped case. The instability criteria of the identified solutions have been determined together with the Fourier series transformation up to the first and high harmonics in the sense of the physical interpretation. An explicit form for the Ax=dx/dt=(x+M)2⋅N non-linear differential operator corresponding to the considered functions has been derived, and its main functional spectrum has been evaluated. Non-isothermal creep tests of different materials were completely described using the Duffing equation via noted solutions up to the fracture as processes with a personal deformation response. We successfully examined the relationship between the thermal and magnetic properties of the ferromagnetic amorphous alloy under its non-linear deformation, using the critical exponents equal to α1 = 2 and α2 = 1. With high linear correlation coefficients (0.9 and above) between our model and experiments (within ±0.01 mm of residual error), the behavior of organic and metallic systems is well predicted under the same thermomechanical testing conditions on the mesoscale.