Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

Abstract

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a "spreading" discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity.We consider specific features of the behavior of nonlinear quasitransverse waves in composite elastic media where the long-wave perturbations are described by nonlinear hyperbolic equations expressing the conservation laws. Discontinuities can appear in the solutions of these equations in their evolution process. Dispersion and dissipation are essential in smaller-scale phenomena. To describe the processes in narrow regions that are replaced with discontinuities for large scales, we use a more complete system of equations taking the large-and small-scale phenomena into account. The discontinuities associated with a continuous traveling-wave-type solution of the complete system of equations (the solution of the problem of finding the discontinuity structure) are usually said to be admissible. The influence of dispersion and dissipation results in the structure of the set of admissible discontinuities becoming rather complicated. We assume that a hyperbolic system of equations with the corresponding system of relations for the discontinuities and with the additional requirement that the discontinuities belong to the set of admissible discontinuities is taken as a model of large-scale phenomena. Then, as was shown in [1]-[3] for nonlinear waves in rods and magnetic solids as well as for a certain model of composite media, the solutions of self-similar problems for some distinguished domains of initial parameters can be nonunique. The number of solutions in these domains depends on the dispersion inside the discontinuity structure (for a constant viscosity) and increases without bound as the dispersion increases. In this paper, we numerically analyze self-similar asymptotic approximations in the nonuniqueness domains as a result of the evolution of nonstationary solutions of the complete system of partial differential equations. A model of large-scale phenomenaIf the medium anisotropy is small, then weakly nonlinear quasitransverse waves propagating in the positive direction of the x axis can be described by a simplified system of equations obtained from the

Classical and nonclassical discontinuities and their structures in nonlinear elastic media with dispersion and dissipation

“…Part of classical discontinuities (shock waves) turn out to be inadmissible. If the model of large-scale phenomena consists of the hyperbolic system of equations of nonlinear elasticity theory, which express conservation laws, and the related constraints on discontinuities supplemented with the requirement that the discontinuities belong to the set of admissible discontinuities, then, as shown in [25,[49][50][51][52] for nonlinear waves in rods and magnets and for a certain model of composite medium, the solutions of self-similar problems are nonunique in a distinguished parameter range. The number of solutions in such a parameter range depends on the relative influence of dispersion inside the structures of discontinuities in comparison with that of viscosity and unboundedly grows with increasing this influence.…”

Классические И Неклассические Разрывы В Решениях Уравнений Нелинейной Теории Упругости

“…Ниже приводятся результаты численного решения задач для уравнений (3.1) при различном задании начальных условий (3.3) [47], [48].…”

Stability of shock wave structures in nonlinear elastic media

“…The discontinuity on which the relations following from the conservation laws, together with the condition of the existence of the structure, are met is called the evolutionary discontinuity. If the discontinuity is not a-priori evolutionary, in addition to the conditions following from the conservation laws, the relations following from the requirement for the structure to exist to make this discontinuity evolutionary are met, this is called a special discontinuity [8][9][10][11][12][13][14][15][16][17]. In addition, the requirement for the structure can make some a-priori evolutionary discontinuities become non-evolutionary.…”

“…The solutions to nonlinear hyperbolic equations representing special discontinuities have been analyzed for various models of continuum mechanics [8][9][10][11][12][13][14][15][16][17], and the behavior of similar solutions which can also be called special discontinuities have been studied [18]. The existence of special discontinuities could lead to the non-uniqueness of the Riemann problem solutions [12][13][14][15][16][17].…”

“…It has been shown in other studies [3][4][5] that the behavior of simple waves and shock waves in nonlinear elastic media without dispersion depends essentially on the sign of the nonlinearity coefficient. The stationary structure of the discontinuity has been analyzed for positive and negative nonlinearity parameters [12][13][14], and it was shown that for the essential dispersion effect the number of different special discontinuity types grows when the dispersion effect grows, as compared with dissipation.…”