Simple waves in nonlinearly elastic media

Свешникова, Е.И.

doi:10.1016/0021-8928(82)90038-7

Cited by 10 publications

(12 citation statements)

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“…In many cases the hypothesis makes it possible to select a unique solution of the problem [8,11]. In particular, this is the case for solutions to Equations (18) when the discontinuity structure is described by (19) with m = 0 [11].…”

Section: Long Waves In Nonlinear Media With Dispersion and Dissipatiomentioning

confidence: 99%

“…Riemann waves in elastic media were considered in [18, 3, 5, Chapter III]. In the case of small-amplitude waves in weakly anisotropic media it follows [15,17,19, Chapter 3; Section 7.4.5] from (1), (7) that…”

Section: Riemann and Shock Waves Solution Of Piston Problemmentioning

confidence: 99%

“…The set of values behind the shock u + for which the solution of the structure problem (19) exists for a given value ahead of the shock u − was investigated in [33,36]. It consist of intervals and separate points.…”

Section: Long Waves In Nonlinear Media With Dispersion and Dissipatiomentioning

confidence: 99%

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On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

2006

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One-dimensional selfsimilar problems for waves in an elastic half-space generated by a sudden change of the boundary stress (the "piston" problem) and problems of disintegration of an arbitrary discontinuity are considered. For the case when small-amplitude waves are generated in a medium with small anisotropy a qualitative analysis shows that these problems have nonunique solutions when it is assumed that the solutions involve Riemann waves and evolutionary discontinuities. The above-mentioned problems are considered as limits of properly formulated problems for visco-elastic media when the viscosity tends to zero or (what is the same) that time tends to infinity. It is numerically found that all above-mentioned inviscid solutions can represent the asymptotics of visco-elastic solutions. The type of asymptotics depends on those details of the visco-elastic problem formulation which are absent when formulating inviscid selfsimilar problems. Similar considerations are made for elastic media with dispersion along with dissipation which are manifested in small-scale processes. In such media the number of available asymptotics (as t → ∞) for the above-mentioned solutions depends on a relation between dispersion and dissipation and can be large. Thus, two possible causes for the nonuniqueness of solutions to the equations of elasticity theory are investigated.

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Section: Long Waves In Nonlinear Media With Dispersion and Dissipatiomentioning

confidence: 99%

Section: Riemann and Shock Waves Solution Of Piston Problemmentioning

confidence: 99%

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On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

2006

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“…Эти решения изучены в рабо-тах [21], [25]. Подставив (1.5) в (1.3), получим систему обыкновенных диффе-ренциальных уравнений для u α :…”

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“…Интегральные кривые волн Римана касаются собственных векторов матрицы R α,β и на плоскости (u 1 , u 2 ) образуют два ортогональных семейства кривых (рис. 1.1) [21], [22], [25]. Оси u 1 и u 2 для множества интегральных кривых являются осями симметрии.…”

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Классические И Неклассические Разрывы В Решениях Уравнений Нелинейной Теории Упругости

Куликовский¹,

Chugainova²

2008

Uspekhi Mat. Nauk

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Investigation of the shock adiabat of quasitransverse shock waves in a prestressed elastic medium

Куликовский¹,

Свешникова²

1982

Journal of Applied Mathematics and Mechanics

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Simple waves in nonlinearly elastic media

Cited by 10 publications

References 0 publications

On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

Классические И Неклассические Разрывы В Решениях Уравнений Нелинейной Теории Упругости

Investigation of the shock adiabat of quasitransverse shock waves in a prestressed elastic medium

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