Localization of two-dimensional non-linear strain waves in a plate

The influence of an external medium on the evolution of two-dimensional long non-linear strain waves in an elastic plate is studied. The governing non-linear equations for longitudinal and shear waves are obtained. A threshold value of the external medium parameter is found that separates the existence of either one-dimensional (or plane) localized strain wave or two-dimensional localized strain wave. A considerable increase in the amplitude of the wave is found during the formation of the two-dimensional localized strain wave from an arbitrary initial pulse.     

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.     

Some remarks on the growth behavior of one-dimensional shock waves in non-linear materials

The growth behavior of both compressive and expansive one-dimensional shock waves which propagate into an unstrained region of a non-linear material exhibiting anelastic response, in the sense of Eckart, is analyzed. In each case, a differential equation governing the growth of the amplitude of the shock is derived and it is shown that a critical strain gradient may be defined. The growth behavior of the waves closely resembles the growth behavior of compressive and expansive shock waves propagating in sufficiently smooth non-linear materials with fading memory, i.e., in materials which can be approximated by linear viscoelastic materials for small relative strains.     

On the effective stiffness of plates made of hyperelastic materials with initial stresses

Within the framework of the direct approach to the plate theory we consider the infinitesimal deformations of a plate made of hyperelastic materials taking into account the non-homogeneously distributed initial stresses. Here we consider the plate as a material surface with 5 degrees of freedom (3 translations and 2 rotations). Starting from the equations of the non-linear elastic body and describing the small deformations superposed on the finite deformation we present the two-dimensional constitutive equations for a plate. The influence of initial stresses in the bulk material on the plate behavior is considered.     

Transport equations for elastic and other waves in random media

We derive and analyze transport equations for the energy density of waves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization effects, coupling of different types of waves, etc. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion coefficients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.     

On Material Conservation Laws for a Consistent Plate Theory

In the paper, material conservation laws associated with a consistent second-order plate theory are derived, which takes shear deformations and strains in thickness direction of the plate into account. Three path-independent integrals are established. In the presence of inhomogeneities in the material (e.g., defects or cracks), energy-release rates due to the change of the configuration of such flaws can be calculated by these integrals. The resulting material forces may serve to assess the reliability of structures with cracks.     

Velocity dispersion in an elastic plate with microstructure: effects of characteristic length in a couple stress model

Microstructural effects become important, when dimensions of the heterogeneous material are comparable to the length scale of microstructure and the state of stress needs to be defined in a non-local manner. Linear theory of elasticity, which is associated with the concept of homogeneity of material and local stresses, cannot describe the behavior of the materials with microstructures. In this study, Couple stress theory of elasticity has been employed to capture the size effects on the propagation of Lamb waves in an elastic plate with microstructure. Effects on the dispersion curves of Lamb waves are studied, when the characteristic length of the material is comparable to cell size. The governing equations of couple stress theory, involving stresses and couple stresses are solved to study the impact of different characteristic lengths, comparable with cell size. Since bone is a material with microstructure, so for numerical calculations and graphical representation of the results, the plate is considered to have mechanical properties typically used for bones.     

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.     

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.     

Novel characteristics of lump and lump–soliton interaction solutions to the generalized variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - With the inhomogeneities of media taken into account, a generalized variable-coefficient Kadomtsev–Petviashvili (vcKP) equation is proposed to model nonlinear waves in...     

Theory of non-linear wave propagation with application to the interaction and inverse problems

The propagation of non-linear deformation waves in a dissipativc medium is described by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves. The mathematical models are derived from the theories of thermoclasticity or viscoclasticity taking into account the geometric and physical non-linearities and dispersion. On the basis of eikonal equations for the associated linear problem the transport equations of the nth order are obtained. In the multidimensional case the method of matched separation of initial equations is proposed. The interaction problems which occur in head-on collisions and in reflection from boundaries or interfaces are analyzed. Conditions are also studied when the interaction of non-linear waves does not take place. The inverse problem of determining materials properties according to pulse shape changes is discussed.     

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.     

Diagnostics of the medium structure by long wave of finite amplitude

The averaged systems of hydrodynamic equations for a structured medium in the Lagrangian and the Eulerian coordinates are discussed. In the general case, the equations cannot be reduced to the average hydrodynamic terms. Under propagation of long waves in media with structure, the non-linear effects appear and they are analyzed in the framework of the asymptotic averaged model. The heterogeneity in a medium structure always increases the non-linear effects for the long-wave perturbations. A new method for diagnostics of the properties of medium components by long non-linear waves is suggested (inverse problem). The mass contents of components in the media can be determined by this diagnostic method.     

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.     

SHEAR-HORIZONTAL WAVES IN A ROTATED Y-CUT QUARTZ PLATE WITH AN ISOTROPIC ELASTIC LAYER OF FINITE THICKNESS

We study shear-horizontal (SH) waves in a rotated Y-cut quartz plate carrying an isotropic elastic layer of finite thickness.The three-dimensional theories of anisotropic elasticity and isotropic elast...     

Study of failure of layered plates made of composites under impact contact loading

In the present paper, we study the strain and failure of a two-layer plate each of whose layers is made of a composite material. The layers have mutually perpendicular directions of fiber reinforcement. The plate is impacted by a rigid hammer. The layer composite material is modeled by anisotropic elastoviscoplastic damageable media according to two different (one-velocity and two-velocity) models. We propose to use the two-scale theory of fracture of the composite material. The problem is solved numerically by the method of spatial characteristics. This method permits correctly satisfying the boundary and contact conditions and correctly taking into account the material anisotropy in the difference approximation. We show that the one-velocity model increases the degree of plate fracture compared with the two-velocity model, which takes the stress waves dispersion into account. This can be explained by the fact that the stress field in the unloading wave spreads owing to the microinhomogeneity of the composite layers.     

Necking of pressurized spherical membranes

The necking of spherical membranes subject to a prescribed increase in enclosed volume is investigated. Attention is restricted to axisymmetric deformations. The materials considered are incompressible, isotropic, time-independent and incrementally linear. A complete set of axisymmetric bifurcation modes is considered and a simple relation is found to govern the critical stress for bifurcation into a given mode. The limiting critical stress and the corresponding mode for short wavelengths are investigated and related to the results obtained from an independent local-necking analysis. Two perturbation methods are employed to study the growth of initial imperfections: one is valid for arbitrary modes, but restricted to small deviations from sphericity, and the other is valid only for the local-necking mode, but is not restricted to small deviations. The effect of path-dependent material behavior on the onset of local necking is explored. Path-dependent material behavior is found to encourage the preferential growth of short wavelength imperfections. Path-independent materials are shown to exhibit significant sensitivity to initial imperfections in the localized-necking mode, although this sensitivity is far less than for a path-dependent material. When account is taken of initial material-property inhomogeneities as well as initial thickness imperfections, it seems that no definite conclusion can be drawn concerning the appropriateness or inappropriateness of an explanation of the onset of localized necking based on a smooth yield-surface plasticity theory and assuming the presence of such initial inhomogeneities.     

Numerical modeling of the receptivity of a supersonic boundary layer to acoustic disturbances

The receptivity of a supersonic (M_∞ = 6) boundary layer on a flat plate to acoustic disturbances is investigated on the basis of a numerical solution of the 2D Navier-Stokes equations. Numerical results obtained for fast and slow acoustic waves impinging on the plate at zero angle agree qualitatively with asymptotic theory. Calculations carried out for other angles of incidence of the acoustic waves reveal new features of the perturbation field in the neighborhood of the leading edge of the plate. It is shown that, due to visco-inviscid interaction, the shock formed near the leading edge may significantly affect the acoustic field and the receptivity.     

Gravitational and acoustic waves due to the motion of an oscillating thin plate beneath the surface of a heavy weakly-compressible fluid

The three-dimensional problem of the motion of a thin plate in an inviscid, heavy, weakly-compressible fluid is solved. The surface tension is disregarded. The plate moves rectilinearly at a constant velocity under the surface of an infinite-depth fluid and oscillates at a given frequency. The fluctuating dipole potential is obtained from the Euler and continuity equations with account for the conditions on the free surface (linear theory of small waves) and the conditions at infinity. The density distribution function of the dipole layer is determined from the boundary conditions imposed on the plate surface. Formulas for calculating the far acoustic field are derived. The calculations for a square plate are carried out.