Selection of two-dimensional nonlinear strain waves in micro-structured mediaSelection d'ondes de déformation non linéaires à deux dimensions dans des milieux à microstructure

It is shown that the micro-displacement gradient allows the propagation of two-dimensional localized long nonlinear strain waves in a medium with microstructure. These waves may exist even in the presence of dissipation and energy input in the microstructured medium but with selected values of the wave amplitude and velocity. An increase or a decrease in the wave amplitude and velocity happens faster at the initial stage than that of the plane localized wave. However, their steady values selected by the energy input/output, are higher for the plane waves. To cite this article: A.V. Porubov et al., C. R. Mecanique 332 (2004).     

Cubic non-linearity and longitudinal surface solitary waves

It is shown that for some seismic media both quadratic and cubic non-linearities should be taken into account in the governing equation for longitudinal waves. The new equation is obtained to account for non-linear surface waves in a medium surrounding a non-linearly elastic rod. Exact solutions of the equation allow us to describe simultaneous propagation of tensile and compressive localized strain waves. Various interactions between these waves give rise to both the multi-bump and “Mexican hat” localized wave structures closer to the surface waves recently observed in experiments.     

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

A multiple monopole （or multipole） method based on the generalized mul- tipole technique （GMT） is proposed to calculate the band structures of scalar waves in two-dimensional phononic crystals which are composed of arbitrarily shaped cylinders embedded in a host medium. In order to find the eigenvalues of the problem, besides the sources used to expand the wave field, an extra monopole source is introduced which acts as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure is obtained. Some numerical examples are presented to validate the proposed method.     

Nonlinear strain wave localization in periodic composites

Nonlinear strain wave propagation along the lamina of a periodic two-component composite was studied. A nonlinear model was developed to describe the strain dynamics. The model asymptotically satisfies the boundary conditions between the lamina, in contrast to previously developed models. Our model reduces an initial two-dimensional problem into a single one-dimensional nonlinear governing equation for longitudinal strains in the form of the Boussinesq equation. The width of the lamina may control the propagation of either compression or tensile localized strain waves, independent of the elastic constants of the materials of the composite.     

Non-linear bell-shaped and kink-shaped strain waves in microstructured solids

The evolution of finite-amplitude strain waves is studied in a medium with microstructure when dissipation and energy input are taken into account. The governing non-linear equation for longitudinal strain waves is obtained in the one-dimensional case. The propagation and attenuation or amplification of bell-shaped and kink-shaped waves, whose parameters are defined in an explicit form through the parameters of the microstructured medium, are studied.     

On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves

The evolution of a three-dimensional packet of weakly non-linear internal gravity waves propagating obliquely at an arbitrary angle to the vertical line is considered. Two coupled non-linear equations connecting variations of a packet amplitude and induced flows are derived. three-dimensionality of the packet having been found responsible for the non-linearity of the system. Explicit formulae for the induced flow vertical component and the mean density field variation caused by packet propagation have been obtained. The plane wave is shown to be unstable at any arbitrary slope of the wave vector. The non-linear equation describing the evolution of the two-dimensional packet is derived in the subsequent order of the asymptotic scheme.It has been found possible for the packet to collapse. The collapse of internal waves packets may be one of the possible mechanisms of “blini”-shaped regions of mixed waters formation in the ocean.     

Non-linear interaction of water waves with submerged obstacles

The interaction of two-dimensional water waves with a fixed submerged cylinder is studied using a finite difference scheme with boundary-fitted co-ordinates. A mixed Eulerian–Lagrangian (MEL) formulation is used to satisfy the fully non-linear free surface conditions. The diffraction of small-amplitude water waves by a cylinder is examined for various wavelengths and amplitudes of the incident wave. Fourier analyses of the incident and diffracted waves are performed to determine their spectra. An example of a large-amplitude wave breaking over a cylinder is also studied. The non-linear numerical solutions are compared with those of experiments and linear theory where appropriate.     

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.     

Band structure calculations of in-plane waves in two-dimensional phononic crystals based on generalized multipole technique

A numerical method, the so-called multiple monopole (MMoP) method, based on the generalized multipole technique (GMT) is proposed to calculate the band structures of in-plane waves in two-dimensional phononic crystals, which are composed of arbitrarily shaped cylinders embedded in a solid host medium. To find the eigenvalues (eigenfrequencies) of the problem, besides the sources used to expand the wave fields, an extra monopole source is introduced which acts as the external excitation. By varying the excitation frequency, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and the boundary of the irreducible first Brillouin zone (FBZ), the band structures can be obtained. Some typical numerical examples with different acoustic impedance ratios and with inclusions of various shapes are presented to validate the proposed method.     

Localized transition waves in bistable-bond lattices

Discrete two-dimensional square- and triangular-cell lattices consisting of point particles connected by bistable bonds are considered. The bonds follow a trimeric piecewise linear force-elongation diagram. Initially, Hooke's law is valid as the first branch of the diagram; then, when the elongation reaches the critical value, the tensile force drops to the other. The latter branch can be parallel with the former (mathematically this case is simpler) or have a different inclination. For a prestressed lattice the dynamic transition is found analytically as a wave localized between two neighboring lines of the lattice particles. The transition wave itself and dissipation waves carrying energy away from the transition front are described. The conditions are determined which allow the transition wave to exist. The transition wave speed as a function of the prestress is found. It is also found that, for the case of the transition leading to an increased tangent modulus of the bond, there exists nondivergent tail waves exponentially localized in a vicinity of the transition line behind the transition front. The previously obtained solutions for crack dynamics in lattices appear now as a partial case corresponding to the second branch having zero resistance. At the same time, the lattice-with-a-moving-crack fundamental solutions are essentially used here in obtaining those for the localized transition waves in the bistable-bond lattices. Steady-state dynamic regimes in infinite elastic and viscoelastic lattices are studied analytically, while numerical simulations are used for the related transient regimes in the square-cell lattice. The numerical simulations confirm the existence of the single-line transition waves and reveal multiple-line waves. The analytical results are compared to the ones obtained for a continuous elastic model and for a related version of one-dimensional Frenkel-Kontorova model.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Non-linear Evolution of P-waves in Viscous–Elastic Granular Saturated Media

The Frenkel–Biot P-wave of the first type is a seismic longitudinal wave observed in rocks fully saturated with oil, water or high-pressure gas. The P-wave of the second type is observed in unsaturated soils and other porous media saturated with gas of low pressure. Their models include properties of the skeleton, that is, its elastic modules and its own viscosity. If the non-linear terms are accounted for, the asymptotic analysis, usual for weak non-linear waves, might be applied to get the wave spectrum evolution. The wetness of grains contacts in soils and such components of oil as tars or bitumen, which attached to the skeleton, can be described by generalized viscous–elastic stress–strain connections. The latter are nominated in such a way that creates the narrow frequency interval of wave of negative dissipation where the non-linear terms begin to play the main role besides the neutral stability for waves of zero wave number. The corresponding case, relevant to single continuum model, was analyzed in the literature. Here it is shown that the interpenetrating continua with interaction of the Darcy type provide the dissipation sink in the wave evolution equation. This generalization, (Tribelsky, M.I.: Phys. Rev. Lett. (2007, submitted)), can stabilize the asymptotic solution of the evolution equation, where the dispersion terms are omitted. The asymptotic solution of the equation is invariant to initial conditions and it means a transformation of initial wave spectra to unique one while wave is spreading in the viscous–elastic medium under consideration. This explains the phenomenon, observed in wave tests at marine beach, when any dynamics action (impact, explosion, and ultrasound action) created at some distance a wave of a single frequency (~25 Hz).     

Diffraction of an elastic wave by an embedded quarter-space

A method of matching asymptotic fields has recently been applied to the problem of the diffraction of a plane time-harmonic acoustic wave by an embedded quarter-space with different acoustic properties from the rest of space. The method is here applied to the equivalent problem of elastic waves. The normal to the incident wavefront is perpendicular to the apex of the quarter-space and so the problem is two-dimensional in plane strain.Exact expressions are found for the far-field on the boundary of the quarter-space, neglecting those terms which decay faster than the inverse half power of the distance. The main case of interest is where the incident wave propagates parallel to one of the interfaces.The method, unfortunately, does not lead to any information about the amplitude of any interface (Stoneley or Rayleigh) wave which may exist.     

Two-dimensional modulation and instabilities of flexural waves of a thin plate on nonlinear elastic foundation

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. It is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear Schrödinger equation. The latter equation allows us to study the modulational instability conditions leading to different zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.     

Parametric identification of crystals having a cubic lattice with negative Poisson’s ratios

A two-dimensional model of an anisotropic crystalline material with cubic symmetry is considered. This model consists of a square lattice of round rigid particles, each possessing two translational and one rotational degree of freedom. Differential equations that describe propagation of elastic and rotational waves in such a medium are derived. A relationship between three groups of parameters is found: second-order elastic constants, acoustic wave velocities, and microstructure parameters. Values of the microstructure parameters of the considered anisotropic material at which its Poisson’s ratios become negative are found.