Damage localization using transient non-linear elastic wave spectroscopy on composite structures

To reduce the costs related to maintenance of aircraft structures, there is the need to develop new robust, accurate and reliable damage detection methods. A possible answer to this problem is offered by newly developed non-linear acoustic/ultrasonic techniques, which monitor the non-linear elastic wave propagation behaviour introduced by damage, to detect its presence and location.In this paper, a new transient non-linear elastic wave spectroscopy (TNEWS) is presented for the detection and localization of a scattered zone (damage) in a composite plate. The TNEWS analyses the uncorrelations between two structural dynamic responses generated by two different pulse excitation amplitudes by using a time-frequency coherence function. A numerical validation of the proposed method is presented. Damage was introduced and modelled using a multi-scale material constitutive model (Preisach-Mayergoyz space).The developed technique identified in a clear manner the faulted zone, showing its robustness to locate and characterize non-linear sources in composite materials     

Low-frequency band gaps in chains with attached non-linear oscillators

The aim of this article is to investigate the wave propagation in one-dimensional chains with attached non-linear local oscillators by using analytical and numerical models. The focus is on the influence of non-linearities on the filtering properties of the chain in the low frequency range. Periodic systems with alternating properties exhibit interesting dynamic characteristics that enable them to act as filters. Waves can propagate along them within specific bands of frequencies called pass bands, and attenuate within bands of frequencies called stop bands or band gaps. Stop bands in structures with periodic or random inclusions are located mainly in the high frequency range, as the wavelength has to be comparable with the distance between the alternating parts. Band gaps may also exist in structures with locally attached oscillators. In the linear case the gap is located around the resonant frequency of the oscillators, and thus a stop band can be created in the lower frequency range. In the case with non-linear oscillators the results show that the position of the band gap can be shifted, and the shift depends on the amplitude and the degree of non-linear behaviour.     

Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load

Vibration of a finite Euler–Bernoulli beam, supported by non-linear viscoelastic foundation traversed by a moving load, is studied and the Galerkin method is used to discretize the non-linear partial differential equation of motion. Subsequently, the solution is obtained for different harmonics using the Multiple Scales Method (MSM) as one of the perturbation techniques. Free vibration of a beam on non-linear foundation is investigated and the effects of damping and non-linear stiffness of the foundation on the responses are examined. Internal-external resonance condition is then stated and the frequency responses of different harmonics are obtained by MSM. Different conditions of the external resonance are studied and a parametric study is carried out for each case. The effects of damping and non-linear stiffness of the foundation as well as the magnitude of the moving load on the frequency responses are investigated. Finally, a thorough local stability analysis is performed on the system.     

Simulation of multi-cracks in solids using nonlinear elastic wave spectroscopy with a time-reversal process

In non-classical nonlinear media, much characteristic information is contained in their dynamic elastic responses. A method combining nonlinear elastic wave spectroscopy (NEWS) with a time-reversal (TR) process is used in this numerical study, in which the presence of one defect and two defects acting with non-classical nonlinearity in an attenuating medium is simulated. Nonlinear defect behavior is introduced using a modified Preisach–Mayergoyz (PM) model. Two methods are used to determine retrofocal quality: harmonic filtering and modulated wave filtering. In the simulation, the nonlinear signal is filtered from the received continuous wave, then reversed and re-sent; a crack image can be obtained from the nonlinear signal in a lossy solid. By comparison with the actual defect, the image can reflect the distribution of one or two flaws, which show the feasibility and value of the NEWS–TR methodology for microdamage imaging of two defects. These results also show that images obtained with different harmonic and modulated frequencies can reflect the presence of defects. With increasing frequency, the crack positions obtained from the image change due to the influence of solid loss and interaction with sound waves.     

Modal interactions in the vibrations of a heated annular plate

We investigate the non-linear forced vibrations of a thermally loaded annular plate with clamped–clamped immovable boundary conditions in the presence of a three-to-one internal resonance between the first and second axisymmetric modes. We consider the in-plane thermal load to be axisymmetric and excite the plate externally by a harmonic force near primary resonance of the second mode. We then use the non-linear von Kármán plate equations to model the behavior of the system and apply the method of multiple scales to investigate its responses. We found that the response can be periodic oscillations consisting of both modes, with a large component from the first mode. Moreover, the periodic solutions may undergo Hopf bifurcations, which lead to aperiodic oscillations of the plate.     

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.     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

垂直入射弹性波在压电体单侧接触界面上的传播特性

弹性波与压电材料接触界面的相互作用问题是工程应用中常见而复杂的问题,入射波足够强会引起界面出现滑移和分离,但滑移和分离的边界未知,边界条件具有非线性特性。通过Fourier分析,将混合边值问题的求解转化为非线性代数方程,利用软件通过迭代修正的方法进行了求解;给出3种状态边界的求解,分析入射波强度、外加应力及电场对界面状态的影响,并对高频谐波的特性进行分析,通过实例对理论推导进行验证,结果显示:入射波强度、外加荷载和电场的大小及摩擦因数均会影响到界面,通过改变这些条件可以控制界面状态,另外检测高频谐波的信号也可以反映界面状态。     

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.     

A complete boundary integral formulation for steady compressible inviscid flows governed by non-linear equations

A complete boundary integral formulation for steady compressible inviscid flows governed by non-linear equations is established by using the specific mass flux as a dependent variable. Thus, the dimensionality of the problem to be solved is reduced by one and the computational mesh to be generated is needed only on the boundary of the domain. It is shown that the boundary integral formulation developed in this paper is equivalent of the results of distributions of the fundamental solutions of the Laplacian operator equation with a different order along the boundaries of the domain. Hence, we have succeeded in establishing the fundamental-solution method for compressible inviscid flows governed by non-linear equations.     

The non-linear,dynamic response of an impulsively loaded circular plate with a central hole

A numerical method, called Direct Analysis, is described and applied to solve the problem of a plate undergoing a large impulsive load. For generality, an expanded, non-linear form of the equations of motion is used and shear correction and rotatory inertia are considered. The wave speeds are calculated from the non-linear equations and appropriate boundary conditions are applied so that reflected waves are included. The results for two types of step loading pulses are presented and compared with previously presented solutions. The response of the plate is discussed and conclusions as to the effects of the non-linearities are given.     

Pseudo-Spectral simulation of 1D nonlinear propagation in heterogeneous elastic media

In this work, a 1D Pseudo-Spectral Time Domain (PSTD) algorithm has been developed for solving elastic wave equation in nonlinear heterogeneous solids using FFTs for calculation of the spatial differential operator on staggered grid. The solver uses a staggered fourth order Adams–Bashforth method, by which stress and particle velocity are updated at alternating half time steps, to integrate forward in time. To circumvent wraparound inherent to FFT-based pseudo-spectral simulation, Convolution Perfectly Matched Layer (CPML) boundary condition has been used to eliminate implementation problems linked in classical PML to the introduction in nonlinear elasticity of a time dependent bulk modulus. Different kinds of nonlinear elastic models (quadratic and cubic nonlinearity, Nazarov hysteretic nonlinearity, bi-modular nonlinearity, PM-Space nonlinearity) have been implemented. The present study will focus on the comparison of nonlinear signature (harmonics generation, shock, frequency shift and attenuation) of these different kinds of nonlinearity for rod resonance, shock wave generation. These results are expected to be useful in helping to determine the predominant nonlinear mechanism in a specific experiment.     

Non-linear dynamic stability of damped Beck's column with variable cross-section

In this paper the non-linear dynamic stability of Beck's column with variable mass and stiffness properties in the presence of damping (both internal and external) is investigated using a complete non-linear dynamic analysis. This approach permits the examination of the global stability of the system in contrast to the static non-linear one, which, though more economical in computational cost, is associated only with the loss of local stability via flutter or divergence. The governing equations describing the dynamic response are derived in terms of the displacements taking also into account the axial deformation, which has a striking influence on the critical load. Since the cross-sectional properties of the beam vary along its axis, the resulting coupled non-linear differential equations have variable coefficients. Their solution is achieved using the analog equation method (AEM) of Katsikadelis. Besides its accuracy and effectiveness, this method overcomes the shortcoming of a FEM solution, which may experience lack of convergence. Interesting conclusions are drawn. The important, however, finding is that the inclusion of the axial deformation affects highly the critical load of Beck's column with varying cross-sectional properties, while it leaves it unaltered for Beck's column with uniform cross-section.     

Symmetry classification and optimal systems of a non-linear wave equation

In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.     

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially non-linear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption.     

Propagation of harmonic waves through micro gap with consideration of frictional contact

Transmission of elastic waves through a micro gap between two solids with consideration of frictional contact is investigated.By using the Fourier analysis technique and the corrective solution method,the nonlinear boundary problem is reduced to a set of algebraic equations.Numerical results exhibit the locations and extents of separation, slip,and stick zones,the interface tractions,and the energy partition.The effects of gap width,frictional coefficients,and the incident angle on the wave transmission ...     

On a Rayleigh Wave Equation with Boundary Damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead transmission lines in a windfield. An asymptotic theory for a class ofinitial-boundary value problems for nonlinear wave equations is presented. Itwill be shown that the problems considered are well-posed for all time t. A multiple time-scales perturbation method incombination with the method of characteristics will be used to construct asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Some numerical results are alsopresented in this paper.     

The application of finite element analysis to the solution of Stokes wave diffraction problems

A new finite-element based method of calculating non-linear wave loads on offshore structures in extreme seas is presented in this paper. The diffraction wave field is modelled using Stokes wave theory developed to second order. Wave loads and free surface elevations are obtained for fixed surface-piercing structures by solving a boundary value problem for the second-order velocity potential. Special attention has been given to the radiation condition for the second-order diffraction field. Results are presented for three test examples, the vertical cylinders of Kim and Yue and of Chakrabarti, and an elliptic cylinder. These results demonstrate that early problems with the application of second-order theory arising from inadequate radiation conditions have been overcome.