Longitudinal strain waves in non-linearly-elastic media with couple stresses

Erofeyev, Vladimir I.; Potapov, A. I.

doi:10.1016/0020-7462(93)90021-c

Cited by 29 publications

(12 citation statements)

References 4 publications

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“…(5.26) Here and furt her , the proper scaling is taken to simplify the evolution equations as much as possible (Engelbrecht, 1983) and indices denote the differentiation. (5.26) as shown by Erofeyev and Potapov (1993). A similar result is obtained in the first approximation for waves in waveguides (rods) to model the influence of boundaries (geometric dispersion) (Engelbrecht, 1983) while more detailed studies show the need to use a more complicated wave equation (Samsonov, 1994) where the splitting-up to single (unidirectional) waves is impossible (see Section 5.2).…”

Section: Korteweg-de-vries Equationsupporting

confidence: 59%

Nonlinear Wave Dynamics

Engelbrecht

1997

Kluwer Texts in the Mathematical Sciences

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Section: Korteweg-de-vries Equationsupporting

confidence: 59%

Nonlinear Wave Dynamics

Engelbrecht

1997

Kluwer Texts in the Mathematical Sciences

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“…Actually, Cosserat's approach can be further generalised including in macroscopic models, along with micro-rotations, also micro-stretches, micro-strains or concentrated micro-deformations, so introducing the so-called micro-structured or micropolar or micromorphic continuum models (Erofeyev, Potapov, 1993;Potapov, Pavlov, Maugin, 1999;Erofeyev, Pavlov, Leontiev, 2013). These can be formulated through a postulation process based on the principle of least action (Auffray et al, 2013) or on the principle of virtual works (Maugin, Metrikine, 2010;Maugin, 2013).…”

Section: Cosserat and Micromorphic Continuamentioning

confidence: 99%

Dynamic problems for metamaterials: Review of existing models and ideas for further research

Vescovo

Giorgio

2014

International Journal of Engineering Science

203

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Metamaterials are materials especially engineered to have a peculiar physical behaviour, to be exploited for some well-specified technological application. In this context we focus on the conception of general microstructured continua, with particular attention to piezoelectromechanical structures, having a strong coupling between macroscopic motion and some internal degrees of freedom, which may be electric or, more generally, related to some micro-motion. An interesting class of problems in this context regards the design of wave-guides aimed to control wave propagation.The description of the state of the art is followed by some hints addressed to describe some possible research developments and in particular to design optimal design techniques for bone reconstruction or systems which may block wave propagation in some frequency ranges, in both linear and non-linear fields.Metamaterials are materials which are designed to have exotic behaviour: the concept has been first conceived for optical devices. Therefore, very often one talks about mechanical metamaterials, when the exotic behaviour is limited to mechanical effects, as e.g. very negative Poisson effects. This paper is based on a really simple idea: construct a bridge between two different cultural environments which address the same relevant problems. Metamaterials are studied and conceived by physicists to tackle problems and applications not yet considered in engineering sciences; at the same time the community of continuum mechanics nearly completely ignores what physicists devise and develop in the same field. This review intends to fill a gap in order to stimulate a parallel development of the these theories and to near not communicating scientific groups.The mathematical formalism chosen thereinafter is that preferred by physicist, like Landau type variational principles, and the treated subjects are chosen, in the opinion of the authors, from those considered nowadays more important by applied mechanicians.The capability of continuum theories to describe the time evolution and the deformation of the micro-structure of complex mechanical systems was recognised in the very first formulations of continuum mechanics, as in the pioneering work by Piola (Piola, 1846). He was lead by stringent physical considerations to introduce higher gradients of displacement field, as necessary independent variables, in the constitutive equation for the deformation energy of continuous media. For a more modern interpretation of this subject refer to (Mindlin, However in a similar period, while Piola was producing his papers, Cauchy and Poisson obtained a description, with a very elegant and effective format, for continuum mechanics in which: i) the displacement from a reference configuration is the only kinematic descriptor;ii) the crucial conceptual tool is Cauchy stress which is constitutively related only to the first gradient of displacement;iii) the crucial postulates are those concerning balance of mass, linear and angular momentum and, when necessary, energy.T...

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“…Thus the resulting equations include higher-order spatial derivatives, representing the contribution of micro deformation, of three unknown functions, i.e., displacement components. The basic equations governing the displacement field of the cubically nonlinear elastic medium whose constitutive equations include higher-order displacement gradients have been given in [14]. The following set of equations describe the one-dimensional motion in the elastic medium:…”

Section: Basic Equations and Dispersion Relationsmentioning

confidence: 99%

Coupled quintic nonlinear Schrödinger equations in a generalized elastic solid

Hacinliyan¹,

Erbay²

2004

J. Phys. A: Math. Gen.

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In the present study, the nonlinear modulation of transverse waves propagating in a cubically nonlinear dispersive elastic medium is studied using a multiscale expansion of wave solutions. It is found that the propagation of quasimonochromatic transverse waves is described by a pair of coupled nonlinear Schrödinger (CNLS) equations. In the process of deriving the amplitude equations, it is observed that for a specific choice of material constants and wavenumber, the coefficient of nonlinear terms becomes zero, and the CNLS equations are no longer valid for describing the behaviour of transverse waves. In order to balance the nonlinear effects with the dispersive effects, by intensifying the nonlinearity, a new perturbation expansion is used near the critical wavenumber. It is found that the long time behaviour of the transverse waves about the critical wavenumber is given by a pair of coupled quintic nonlinear Schrödinger (CQNLS) equations. In the absence of one of the transverse waves, the CQNLS equations reduce to the single quintic nonlinear Schrödinger (QNLS) equation which has already been obtained in the context of water waves. By using a modified form of the so-called tanh method, some travelling wave solutions of the CQNLS equations are presented.

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Longitudinal strain waves in non-linearly-elastic media with couple stresses

Cited by 29 publications

References 4 publications

Nonlinear Wave Dynamics

Nonlinear Wave Dynamics

Dynamic problems for metamaterials: Review of existing models and ideas for further research

Coupled quintic nonlinear Schrödinger equations in a generalized elastic solid

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