Spatial and temporal evolution of one-dimensional nonlinear deformation waves in cylindrical shells
DOI:
https://doi.org/10.32347/2410-2547.2025.115.304-314Keywords:
evolution, one-dimensional nonlinear deformation waves, cylindrical shells, cnoidal waves, solitons, elasticity, KdV equationAbstract
In this paper, the physical-mechanical and mathematical models designed to analyze nonlinear wave formations in cylindrical elastic shells are substantiated. Using the perturbation reduction method, the spatial and temporal evolution of nonlinear longitudinal and longitudinal-shift waves in elastic cylindrical shells in the framework of Kirchhoff-Liave theory is investigated. It is established that in elastic cylindrical shells (in one-dimensional formulation of the problem) there exist one-dimensional solitons and nonlinear periodic waves (of the cnoidal type), which are partial solutions of the Korteweg-de Vries equation (KdV). All the main parameters of these types of waveforms are analytically found. Taking into account dissipative effects, the evolutionary Korteweg-de Vries-Bürgers equation (KDB), which is close to the integrated ones, was obtained for the longitudinal deformation component. Partial solutions of this equation in the form of kinks and waveforms, which are described by the elliptic Weierstrass function, have been found analytically (using the reduced expansion method). The analysis performed allows us to describe the spatial and temporal evolution of a weakly two-dimensional beam of nonlinear longitudinal and longitudinal-shift waves in an elastic cylindrical shell as follows. There is a fast (linear) time scale during which there exists a wave running without profile change with constant velocity.
Further, there is a slower time scale, during which the change of wave parameters due to nonlinearity, dispersion, and diffraction divergence leads to the splitting of the initial pulse into one-dimensional (or weakly two-dimensional) solitons or the formation of cnoidal (periodic) waves. The interaction of longitudinal deformation waves unstable to transverse perturbations with longitudinal-shift waves (solitons/cnoidal waves) leads to their overturning and, possibly, to their destruction. This circumstance essentially distinguishes the spatial and temporal evolution of solitons in deformed solids from hydrodynamic solitons, which collapse gradually due to natural dissipation.
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