Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories

Long-wave models obtained in the process of asymptotic homogenisation of structures with a characteristic length scale are known to be non-unique. The term non-uniqueness is used here in the sense that various homogenisation strategies may lead to distinct governing equations that usually, for a given order of the governing equation, approximate the original problem with the same asymptotic accuracy. A constructive procedure presented in this paper generates a class of asymptotically equivalent long-wave models from an original homogenised theory. The described non-uniqueness manifests itself in the occurrence of additional parameters characterising the model. A simple problem of long-wave propagation in a regular one-dimensional lattice structure is used to illustrate important criteria for selecting these parameters. The procedure is then applied to derive a class of continuum theories for a two-dimensional square array of particles. Applications to asymptotic structural theories are also discussed. In particular, we demonstrate how to improve the governing equation for the Rayleigh-Love rod and explain the reasons for the well-known numerical accuracy of the Mindlin plate theory

On rational boundary conditions for higher-order long-wave models

Higher-order corrections to classical long-wave theories enable simple and efficient modelling of the onset of wave dispersion and size effects produced by underlying micro-structure. Since such models feature higher spatial derivatives, one needs to formulate additional boundary conditions when confined to bounded domains. There is a certain controversy associated with these boundary conditions, because it does not seem possible to justify their choice by purely physical considerations. In this paper an asymptotic model for onedimensional chain of particles is chosen as an exemplary higher-order theory. We demonstrate how the presence of higher-order derivative terms results in the existence of non-physical “extraneous” boundary layer-type solutions and argue that the additional boundary conditions should generally be formulated to eliminate the contribution of these boundary layers into the averaged solution. Several new methods of deriving additional boundary conditions are presented for essential boundary. The results are illustrated by numerical examples featuring comparisons with an exact solution for the finite chain

Four simplified gradient elasticity models for the simulation of dispersive wave propagation

Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classical elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretization tools can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality according to Einstein and their behavior in simple initial/boundary value problems

The influence of contact relaxation on underwater noise emission and seabed vibrations due to offshore vibratory pile installation

The growing interest in offshore wind leads to an increasing number of wind farms planned to be constructed in the coming years. Installation of these piles often causes high underwater noise levels that harm aquatic life. State-of-the-art models have problems predicting the noise and seabed vibrations from vibratory pile driving. A significant reason for that is the modeling of the sediment and its interaction with the driven pile. In principle, linear vibroacoustic models assume perfect contact between pile and soil, i.e., no pile slip. In this study, this pile-soil interface condition is relaxed, and a slip condition is implemented that allows vertical motion of the pile relative to the soil. First, a model is developed which employs contact spring elements between the pile and the soil, allowing the former to move relative to the latter in the vertical direction. The developed model is then verified against a finite element software. Second, a parametric study is conducted to investigate the effect of the interface conditions on the emitted wave field. The results show that the noise generation mechanism depends strongly on the interface conditions. Third, this study concludes that models developed to predict noise emission from impact pile driving are not directly suitable for vibratory pile driving since the pile-soil interaction becomes essential for noise generation in the latter case

Wavelet approach to vibratory analysis of surface due to a load moving in the layer

This article is available open access through the publisher’s website at the link below. Copyright @ 2007 Elsevier B.V.The paper analyses theoretically the surface vibration induced by a point load moving uniformly along a infinitely long beam embedded in a two-dimensional viscoelastic layer. The beam is placed parallel to the traction-free surface and the layer under the beam is assumed to be a half space. The response due to a harmonically varying load is investigated for different load frequencies. The influence of the layer damping and moving load speed on the level of vibrations at the surface is analysed and analytical closed form solutions in the integral form for the displacement amplitude and the amplitude spectra are derived. Approximate displacement values depending on Young’s modulus and mass density of layers are obtained. The mathematical model is described by the Euler–Bernoulli beam equation, Navier’s elastodynamic equation of motion for the elastic medium and appropriate boundary and continuity conditions. A special approximation method based on the wavelet theory is used for calculation of the displacements at the surface

A new multi-scale dispersive gradient elasticity modelwith micro-inertia: Formulation and C0-finiteelement implementation

Motivated by nano-scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi-scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro-inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain-gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro-inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth-order equations are rewritten in two sets of symmetric second-order equations so that C0-continuity is sufficient in the finite element implementation. Two sets of unknowns are identified as the microstructural and macrostructural displacements, thus highlighting the multi-scale nature of the present formulation. The associated energy functionals and variationally consistent boundary conditions are presented, after which the finite element equations are derived. Considerable improvements over previous gradient models are observed as confirmed by two numerical examples

Higher-order gradient continuum modelling of periodic lattice materials

This is the author’s version of a work that was accepted for publication in the Computational Materials Science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published at: http://dx.doi.org/10.1016/j.commatsci.2011.05.025The dynamic behaviour of periodic lattice materials is investigated using an equivalent higher-order continuum model obtained by homogenisation of the equations of motion. A gradient continuum enriched with higher-order inertia terms is developed using a combination of finite element discretisation of the unit cell and the continualisation approach. The analysis of the dispersion relations shows that the proposed model is able to capture correctly the physical phenomenon of wave dispersion in lattice structures which is overlooked by classical continuum theories

Commentary on Discussion of ‘On the theory of standing waves in tyres at high vehicle speeds’ by V.V. Krylov and O. Gilbert, Journal of Sound and Vibration 329 (2010) 4398–4408

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Sound and Vibration. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published at: http://dx.doi.org/10.1016/j.jsv.2013.08.03

Vortex-induced vibrations of a freely vibrating cylinder near a plane boundary: experimental investigation and theoretical modelling

This work reports on experiments that were performed with a freely vibrating cylinder exposed to currents and placed near a plane boundary parallel to the cylinder axis. It is observed that the proximity of the boundary affects the vertical response of the cylinder in two ways: (i) for gaps between 0.75 and 2 diameters (D), the amplitude of oscillation is reduced; (ii) for gaps smaller than 0.75D, the cylinder impacts the boundary, resulting in an increase of amplitudes and frequencies of oscillations as the flow is accelerated. The in-line force acting on the cylinder is also examined, and the dependency of its harmonic components on the flow velocity and distance to the boundary is evaluated. Besides the typical amplification of the mean component inside the lock-in region, it is also observed that as the cylinder is placed closer to the boundary, the harmonic component with the frequency of the vertical oscillations increases, while the component with twice that frequency decreases in similar amount. Based on the experimental observations, an existing wake-oscillator model for vortex-induced vibrations is enhanced in order to account for the effect of the boundary. The proposed model introduces an effective damper that is activated when the cylinder reaches a certain distance from the boundary, and a damper/spring set representing the rigidity of the boundary and the dissipation of energy due to impact