The interaction of non-linear waves in a slightly anisotropic viscoelastic medium

Chugainova, A. P.

doi:10.1016/0021-8928(93)90067-v

Cited by 7 publications

(6 citation statements)

References 2 publications

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“…The results on the interaction of a metastable shock wave with background inhomogeneities presented above agree with results on the formation of solutions of the first and second types and on the interaction of nonlinear waves with each other obtained in [40,41,43,44] (see Subsections 2.2-2.3). A mere perturbation of the wave structure (e.g., by varying the viscosity) is insufficient for the decomposition of a metastable wave; it is required to achieve the formation of a second-type solution and eliminate the possibility of the existence of first-type solutions by means of special perturbations.…”

Section: Stability Of Structures Of Shock Waves In a Viscoelastic Medsupporting

confidence: 86%

“…The study of the interaction between two shock waves of different types moving toward each other showed that, in all cases considered in [40,41,43], the solution approached the self-similar asymptotics corresponding to the system of waves in a self-similar solution of the first type.…”

Section: The Interaction Of Nonlinear Waves In a Weakly Anisotropic Mmentioning

confidence: 95%

“…The interaction of shock waves was studied in [40,41,43]. In this section, we consider Example 1, which demonstrates the solution of the problem on the interaction of waves moving in the same direction, and Examples 2-5, Example 2.1.…”

Section: The Interaction Of Nonlinear Waves In a Weakly Anisotropic Mmentioning

confidence: 99%

“…In [40,41,43,44], a Voight viscoelastic medium was considered and a series of non-self-similar problems with asymptotics corresponding to the solution of the self-similar nonviscous problem were numerically solved. It turned out that, depending on the specification of the initial or boundary conditions in the non-self-similar problem, its solution in time or spatial intervals proportional to the viscosity may approach a nonviscous asymptotics as t → ∞ represented by solutions of both the first and the second type.…”

Section: Stability Of Structures Of Shock Waves In a Viscoelastic Medmentioning

confidence: 99%

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

Куликовский

Chugainova

2012

Proc. Steklov Inst. Math.

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PREFACEThis paper studies problems related to the propagation of one-dimensional nonlinear waves in elastic media by analytic and numerical methods. The equations of nonlinear elasticity theory are classified as hyperbolic system expressing conservation laws. Their solutions can be uniquely constructed only when the equations are supplemented with terms making it possible to adequately describe real smallscale phenomena, in particular, the structure of arising discontinuities. We consider the behavior of nonlinear waves in two cases, where the small-scale processes are caused by viscosity alone and where an important role is also played by dispersion. Solutions in these cases differ drastically. In cases with dispersion the complicated behavior of solutions containing discontinuities is described, which qualitatively differs from the behavior of nonlinear waves in cases without dispersion. The revealed special features of the behavior of solutions are not related to any specifics of the equations of nonlinear elasticity theory and are generic for systems of partial differential equations with complex hyperbolic part.

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Section: Stability Of Structures Of Shock Waves In a Viscoelastic Medsupporting

confidence: 86%

Section: The Interaction Of Nonlinear Waves In a Weakly Anisotropic Mmentioning

confidence: 95%

Section: The Interaction Of Nonlinear Waves In a Weakly Anisotropic Mmentioning

confidence: 99%

Section: Stability Of Structures Of Shock Waves In a Viscoelastic Medmentioning

confidence: 99%

See 2 more Smart Citations

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

Куликовский

Chugainova

2012

Proc. Steklov Inst. Math.

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“…The problem of two-shock-structure interaction has been numerically analyzed [29]. In the case of nonuniqueness more simple asymptotics emerge.…”

Section: Waves In Viscous-elastic Media Inviscid Limitmentioning

confidence: 99%

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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Conditions for the decay of a wave into a system of waves in a viscoelastic medium with a negative nonlinearity parameter

Chugainova,

Polekhina

2024

International Journal of Non-Linear Mechanics

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The interaction of non-linear waves in a slightly anisotropic viscoelastic medium

Cited by 7 publications

References 2 publications

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

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

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

Conditions for the decay of a wave into a system of waves in a viscoelastic medium with a negative nonlinearity parameter

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