Wave propagation in periodic buckled beams. Part I: Analytical models and numerical simulations

Periodic buckled beams possess a geometrically nonlinear, load–deformation relationship and intrinsic length scales such that stable, nonlinear waves are possible. Modeling buckled beams as a chain of masses and nonlinear springs which account for transverse and coupling effects, homogenization of the discretized system leads to the Boussinesq equation. Since the sign of the dispersive and nonlinear terms depends on the level of buckling and support type (guided or pinned), compressive supersonic, rarefaction supersonic, compressive subsonic and rarefaction subsonic solitary waves are predicted, and their existence is validated using finite element simulations of the structure. Large dynamic deformations, which cannot be approximated with a polynomial of degree two, lead to strongly nonlinear equations for which closed-form solutions are proposed.     

On the nonlinear interactions and existence of breathers in a chain of coupled pendulums

The paper considers a chain of linearly coupled pendulums. Continues first order system equations are treated via time and space multiple scale method which lead to nonlinear Schrödinger equation. Further investigations on the nonlinear Schrödinger equation detects systems responses in the form of propagated nonlinear waves as functions of their envelope and phases. This provides information about localization of nonlinear waves and their directions in space and time.     

Design and analysis of nonlinear-transformation-based broadband cloaking for acoustic wave propagation

A coordinate-transformation method can be used to design invisibility cloaks for many types of waves, including acoustic waves. The traditional method for designing a cloak depends on a transformation from a virtual space to a physical space. Previous acoustic cloaks that are mainly designed with linear-transformation-based acoustics have drawbacks that acoustic wave trajectories in the cloaks cannot be controlled and tuned. This work uses a nonlinear mapping from a ray trajectory perspective to construct acoustic cloaks with tunable non-singular material properties. Use of a ray trajectory equation is a straightforward and alternate way to study propagation characteristics of different types of waves, which allows more flexibility in controlling the waves. A broadband cylindrical cloak for acoustic waves in an inviscid fluid is realized with layered non-singular, homogeneous, and isotropic materials based on a nonlinear transformation. Some advantages and improvements of the invisibility nonlinear-transformation cloak over a traditional linear-transformation cloak are analyzed. The invisibility capability of the nonlinear-transformation cloak can be tuned by adjusting a design parameter that is shown to have influence on the acoustic wave energy flowing into the region inside the cloak. Numerical examples show that the nonlinear-transformation cloak is more effective for making a domain undetectable by acoustic waves in an inviscid fluid and shielding acoustic waves from outside the cloak than the linear-transformation cloak in a broad frequency range. The methodology developed here can be used to design nonlinear-transformation cloaks for other types of waves.     

Orbital stability of periodic waves for the nonlinear Schrödinger equation

The nonlinear Schr?dinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.     

Transversely isotropic sequentially laminated composites in finite elasticity

A general expression for the energy-density function of sequentially laminated composites is derived. For the class of neo-Hookean composites in the limit of small deformations well-known results for linear transversely isotropic composites are recovered. However, it is shown that under large deformations these composites are not isotropic. Transversely isotropic composites are obtained with sequentially-coated composites in which the next rank composite is constructed by lamination of the previous composite with thin layers of the matrix phase. The transverse behavior of this sequentially-coated composite is neo-Hookean with shear modulus in the form of the Hashin-Shtrikman bounds for the corresponding class of linear composites. Comparison of the behaviors of these composites with recent estimates for transversely isotropic composites reveals good agreement up to relatively large deformations and volume fractions of the inclusion phase.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

Solving the nonlinear periodic wave problems with the Homotopy Analysis Method

An analytic technique, namely the Homotopy Analysis Method (HAM), is applied to solve the nonlinear mKdV equation. Solutions for periodic waves are given and compared with the exact ones, which shows the validity of the HAM for the nonlinear periodic wave problems.     

Characterizing JONSWAP rogue waves and their statistics via inverse spectral data

Rogue waves in random sea states modeled by the JONSWAP power spectrum are high amplitude waves arising over non-uniform backgrounds that cannot be viewed as small amplitude modulations of Stokes waves. In the context of Nonlinear Schrödinger (NLS) models for waves in deep water, this poses the challenge of identifying appropriate analytical solutions for JONSWAP rogue waves, investigating possible mechanisms for their formation, and examining the validity of the NLS models in these more realistic settings. In this work we investigate JONSWAP rogue waves using the inverse spectral theory of the periodic NLS equation for moderate values of the period. For typical JONSWAP initial data, numerical experiments show that the developing sea state is well approximated by the first few dominant modes of the nonlinear spectrum and can be described in terms of a 2- or 3-phase periodic NLS solution. As for the case of uniform backgrounds, proximity to instabilities of the underlying 2-phase solution appears to be the main predictor of rogue wave occurrence, suggesting that the modulational instability of 2-phase solutions of the NLS is a main mechanism for rogue wave formation and that heteroclinic orbits of unstable 2-phase solutions are plausible models of JONSWAP rogue waves. To support this claim, we correlate the maximum wave strength as well as the higher statistical moments with elements of the nonlinear spectrum. The result is a diagnostic tool widely applicable to both model or field data for predicting the likelihood of rogue waves. Finally, we examine the validity of NLS models for JONSWAP data, and show that NLS solutions with JONSWAP initial data are described by non-Gaussian statistics, in agreement with the TOPEX field studies of sea surface height variability.     

Numerical and asymptotic solutions of generalised Burgers’ equation

The generalised Burgers’ equation models the nonlinear evolution of acoustic disturbances subject to thermoviscous dissipation. When thermoviscous effects are small, asymptotic analysis predicts the development of a narrow shock region, which widens, leading eventually to a shock-free linear decay regime. The exact nature of the evolution differs subtly depending upon whether plane waves are considered, or cylindrical or spherical spreading waves. This paper focuses on the differences in asymptotic shock structure and validates the asymptotic predictions by comparison with numerical solutions. Precise expressions for the shock width and shock location are also obtained.     

Optimal control strategies for stochastically excited quasi partially integrable Hamiltonian systems

In this paper two different control strategies designed to alleviate the response of quasi partially integrable Hamiltonian systems subjected to stochastic excitation are proposed. First, by using the stochastic averaging method for quasi partially integrable Hamiltonian systems, an n-DOF controlled quasi partially integrable Hamiltonian system with stochastic excitation is converted into a set of partially averaged Itô stochastic differential equations. Then, the dynamical programming equation associated with the partially averaged Itô equations is formulated by applying the stochastic dynamical programming principle. In the first control strategy, the optimal control law is derived from the dynamical programming equation and the control constraints without solving the dynamical programming equation. In the second control strategy, the optimal control law is obtained by solving the dynamical programming equation. Finally, both the responses of controlled and uncontrolled systems are predicted through solving the Fokker-Plank-Kolmogorov equation associated with fully averaged Itô equations. An example is worked out to illustrate the application and effectiveness of the two proposed control strategies.     

Breather-like director reorientations in a nematic liquid crystal with nonlocal nonlinearity

The nature of nonlinear molecular deformations in a homeotropically aligned nematic liquid crystal (NLC) is presented. We start from the basic dynamical equation for the director axis of a NLC with elastic deformations and adopt space curve mapping procedure to analyze the dynamics. The NLC is governed by an integro-differential perturbed nonlocal nonlinear Schrödinger equation and we solve the same using Jacobi elliptic function method aided with symbolic computation and construct an exact solitary wave solution. In order to better understand the effect of nonlocality on the director reorientations of nematic liquid crystal, we have constructed the component forms of director axis using Darboux vector transformation. This intriguing property as a result of the relation between the coherence of the breather-like solitary deformation and the nonlocality reveals a strong need for a deeper understanding in the theory of self-localization in NLC systems.     

Nonlinear Plane Waves in Finite Deformable Infinite Mooney Elastic Materials

For infinite perfectly elastic Mooney materials, nonlinear plane waves are examined in both two and three dimensions. In two dimensions, longitudinal and shear plane waves are examined, while in three dimensions, longitudinal and torsional plane waves are considered. These exact dynamic deformations, applying to the incompressible perfectly elastic Mooney material, can be viewed as extensions of the corresponding static deformations first derived by Adkins [1] and Klingbeil and Shield [2]. Furthermore, the Mooney strain-energy function is the most general material admitting nontrivial dynamic deformations of this type. For two dimensions the determination of plane wave solutions reduces to elementary mathematical analysis, while in three dimensions an integral of the governing system of highly nonlinear ordinary differential equations is determined. In the latter case, solutions corresponding to particular parameter values are shown graphically. This revised version was published online in June 2006 with corrections to the Cover Date.     

An investigation into electromagnetic force models: differences in global and local effects demonstrated by selected problems

This study investigates the implications of various electromagnetic force models in macroscopic situations. There is an ongoing academic discussion which model is “correct,” i.e., generally applicable. Often, gedankenexperiments with light waves or photons are used in order to motivate certain models. In this work, three problems with bodies at the macroscopic scale are used for computing theoretical model-dependent predictions. Two aspects are considered, total forces between bodies and local deformations. By comparing with experimental data, insight is gained regarding the applicability of the models. First, the total force between two cylindrical magnets is computed. Then a spherical magnetostriction problem is considered to show different deformation predictions. As a third example focusing on local deformations, a droplet of silicone oil in castor oil is considered, placed in a homogeneous electric field. By using experimental data, some conclusions are drawn and further work is motivated.     

Large deformations of nonlinear viscoelastic and multi-responsive beams

This study presents analyses of deformations in nonlinear viscoelastic beams that experience large displacements and rotations due to mechanical, thermal, and electrical stimuli. The studied beams are relatively thin so that the effect of the transverse shear deformation is neglected, and the stretch along the transverse axis of the beams is also ignored. It is assumed that the plane that is perpendicular to the longitudinal axis of the undeformed beam remains plane during the deformations. The nonlinear kinematics of the finite strain beam theory presented by Reissner [27] is adopted, and a nonlinear viscoelastic constitutive relation based on a quasi-linear viscoelastic (QLV) model is considered for the beams. Deformation in beams due to mechanical, thermal, and electric field inputs are incorporated through the use of time integral functions, by separating the time-dependent function and nonlinear measures of field variables. The nonlinear measures are formulated by including higher order terms of the field variables, i.e. strain, temperature, and electric field. Responses of beams under mechanical, thermal, and electrical stimuli are illustrated and the effects of nonlinear constitutive relations on the overall deformations of the beams are highlighted.     

Finite element computation of torsional plastic waves in a thin-walled tube

Summary  A finite element technique is presented for the analysis of one-dimensional torsional plastic waves in a thin-walled tube. Three different nonlinear consitutive relations deduced from elementary mechanical models are used to describe the shear stress–strain characteristics of the tube material at high rates of strain. The resulting incremental equations of torsional motion for the tube are solved by applying a direct numerical integration technique in conjunction with the constitutive relations. The finite element solutions for torsional plastic waves in a long copper tube subjected to an imposed angular velocity at one end are given, and a comparison with available experimental results to assess the accuracy of the constitutive relations considered is conducted. It is demonstrated that the strain-rate dependent solutions show a better agreement with the experimental results than the strain-rate independent solutions. The limitations of the constitutive equations are discussed, and some modifications are suggested. Received 9 February 1999; accepted for publication 28 March 2000     

Nonlinear Dynamics of Shells: Theory,Finite Element Formulation,and Integration Schemes

The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories. A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to. The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations. A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered.     

Low-frequency wave propagation in post-buckled structures

Nonlinear wave propagation in solids and material structures provides a physical basis to derive nonlinear canonical equations which govern disparate phenomena such as vortex filaments, plasma waves, and traveling loops. Nonlinear waves in solids however remain a challenging proposition since nonlinearity is often associated with irreversible processes, such as plastic deformations. Finite deformations, also a source of nonlinearity, may be reversible as for hyperelastic materials. In this work, we consider geometric bucking as a source of reversible nonlinear behavior. Namely, we investigate wave propagation in initially compressed and post-buckled structures with linear-elastic material behavior. Such structures present both intrinsic dispersion, due to buckling wavelengths, and nonlinear behavior. We find that dispersion is strongly dependent on pre-compression and we compute waves with a dispersive front or tail. In the case of post-buckled structures with large initial pre-compression, we find that wave propagation is well described by the KdV equation. We employ finite-element, difference-differential, and analytical models to support our conclusions.     

Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems

A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems (multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The final dynamical programming equations for these control problems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynamical programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illustrate the application and effectiveness of the proposed control strategy.     

Liquid sloshing in a horizontally forced vessel with bottom topography

This paper presents a numerical study of the free-surface evolution for inviscid, incompressible, irrotational, horizontally forced sloshing in a two-dimensional rectangular vessel with an inhomogeneous bottom topography. The numerical scheme uses a time-dependent conformal mapping to map the physical fluid domain to a rectangle in the computational domain with a time-dependent aspect ratio Q(t), known as the conformal modulus. The advantage of this approach over conventional potential flow solvers is the solution automatically satisfies Laplace's equation for all time, hence only the integration of the two free-surface boundary conditions is required. This makes the scheme computationally fast, and as grid points are required only along the free-surface, high resolution simulations can be performed which allows for simulations for mean fluid depths close to the shallow water water regime. The scheme is robust and can simulate both resonate and non-resonate cases, where in the former, the large amplitude waves are well predicted.Results of nonlinear simulations are presented in the case of non-breaking waves for both an asymmetrical ‘step’ and a symmetric ‘hump’ bottom topography. The natural free-sloshing mode frequencies are compared with the small topography asymptotic results of Faltinsen and Timokha (2009) (Sloshing, Cambridge University Press (Cambridge)), and are found to be lower than this asymptotic prediction for moderate and large topography magnitudes. For forced periodic oscillations it is shown that the hump profile is the most effective topography for minimizing the nonlinear response of the fluid, and hence this topography would reduce the stresses on the vessel walls generated by the fluid. Results also show that varying the width of the step or hump has a less significant effect than varying its magnitude.     

Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods

In this paper, we study an integrable nonlinear evolution equation which arises in the context of nonlinear dispersive waves in hyperelastic rods. To consider bounded travelling-wave solutions, we conduct a phase plane analysis. A new feature is that there is a vertical singular line in the phase plane. By considering equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. The trajectories which represent bounded travelling-wave solutions are studied one by one. In total, we find there are 12 types of bounded travelling waves, both supersonic and subsonic. While in literature solutions for only two types of travelling waves are known, here we provide explicit solution expressions for all 12 types of travelling waves. Also, it is noted for the first time that peakons can have applications in a real physical problem.