Self-similar asymptotics of wave problems and the structures of non-classical discontinuities in non-linearly elastic media with dispersion and dissipation

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

“…§ 3. Асимптотическое поведение нелинейных волн в упругих средах с дисперсией и диссипацией В этом параграфе, следуя работам [45]- [48], мы рассматриваем нелинейные волны в среде, описываемой системой уравнений, в которой члены низшего порядка дифференцирования образуют, как и в предыдущем параграфе, ги-перболическую систему нелинейной теории упругости. Члены с производными высшего порядка помимо вязкости, которая учитывалась в предыдущем па-раграфе, описывают также дисперсию.…”

“…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].…”