A.V. Metrikine

One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 1: Generic formulation

This paper is the first in a series of two that focus on gradient elasticity models derived from a discrete microstructure. In this first paper, a new continualization method is proposed in which each higher-order stiffness term is accompanied by a higher-order inertia term. As such, the resulting models are dynamically consistent. A new parameter is introduced that accounts for the nonlocal interaction between variables of the discrete model and of the continuous model. When this parameter is set to proper values, physically realistic behavior is obtained in statics as well as in dynamics. In this sense, the proposed methodology is superior to earlier approaches to derive gradient elasticity models, in which anomalies in the dynamic behavior have been found. A generic formulation of field equations and boundary conditions is given based on Hamiltons principle. In the second paper, analytical and numerical results of static and dynamic response of the second-order model and the fourth-order model will be treated.

Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models

A comparison is made between two-dimensional elastic discrete lattices and a corresponding Cosserat continuum. Firstly, it is demonstrated how the equations of motion of the Cosserat model can be retrieved from those of a lattice model. For this purpose, two lattice geometries are elaborated: a 7-cell hexagonal lattice and a 9-cell square lattice. For both lattices, the individual cells have two translational spring interactions and a rotational spring interaction with their neighbouring cells. Secondly, the dispersion relations for the lattice models and the Cosserat continuum model are examined in order to determine up to which wavelength of the deformation field the Cosserat model accurately represents the underlying discrete micro-structure. The effect of the lattice anisotropy and inhomogeneity on this accuracy is also discussed.

Periodically supported beam on a visco-elastic layer as a model for dynamic analysis of a high-speed railway track

This paper presents a theoretical study of the steady state dynamic response of a railway track to a moving train. The model for the railway track consists of two beams on periodically positioned supports that are mounted on a visco-elastic 3D layer. The beams, supports, and layer are employed to model the rails, sleepers and soil, respectively. The axle loading of the train is modeled by point loads that move on the beams. A method is presented that allows to obtain an expression for the steady-state deflection of the rails in a closed form. On the basis of this expression, the vertical deflection of the rails and its dependence on the velocity of the train is analyzed. Critical velocities of the train are determined and the effect of the material damping in the sub-soil and in the pads on the track response at these critical velocities is studied. The effect of the periodic inhomogeneity of the track introduced by the sleepers is studied by comparing the dynamic response of the model at hand to that of a homogenized model, in which the supports are assumed to be not discrete but uniformly distributed along the track. It is shown that the vertical deflection of the rails predicted by these models resemble almost perfectly. The elastic drag experienced by a high-speed train due to excitation of track vibrations is studied. Considering a French TGV as an example, this drag is calculated using both the inhomogeneous and homogenized models of the track and then compared to the rolling and aerodynamic drag.

One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 2: Static and dynamic response

This paper is the second in a series of two that focus on gradient elasticity models retrieved from a discrete mictrostructure. In the first paper, the governing equations for a second-order model and a fourth-order model have been derived. In this paper, the proposed models are studied by means of static and dynamic examples, both from an analytical point of view and a numerical point of view. Details on the spatial discretization are provided. Finally, an experiment is proposed by which the newly introduced parameter can be determined that is responsible for the nonlocal relation between the continuous and the discrete field variables.

Vibration of a periodically supported beam on an elastic half-space

Abstract The steady-state vibration of a periodically supported beam on an elastic half-space under a uniformly moving harmonically varying load is investigated. The concept of equivalent stiffness of a half-space is used for problem analysis. It is shown that the half-space can be replaced by a set of identical springs placed under each support of the beam. The equivalent stiffness of these springs is a function of the frequency of the beam vibrations and of the phase shift of vibrations of neighboring supports. It is found that the equivalent stiffness is equal to zero for some relationship between the frequency and the phase shift. The reason for this is that the surface waves generated by all supports can come to any support in phase, providing an infinite displacement. It is demonstrated that the equivalent stiffness has a real and an imaginary part. The imaginary part arises due to radiation of waves in the half-space. The expressions are derived for the steady-state response of the beam to the moving load. The limiting case of a constant load is considered, showing that the load moving with the Rayleigh wave velocity causes resonance in the system.

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.

Critical Velocities of a Harmonic Load Moving Uniformly Along an Elastic Layer

The critical (resonance) velocities of a harmonically varying point load moving uniformly along an elastic layer are determined as a function of the load frequency. It is shown that resonance occurs when the velocity of the load is equal to the group velocity of the waves generated by the load. The critical depths of the layer are determined as function of the load velocity in the case the load frequency is proportional to the load velocity. This is of importance for high-speed trains where the loading frequency of the train wheel excitations is mainly determined by the ratio between the train velocity and the distance between the sleepers (ties). It is shown that the critical depths are decreasing with increasing train velocity. It is concluded that the higher the train velocity, the more important are the properties of the ballast and the border between the ballast and the substrate.

Instability of vibrations of a mass that moves uniformly along a beam on a periodically inhomogeneous foundation

Abstract The stability of vibrations of a mass that moves uniformly along an Euler–Bernoulli beam on a periodically inhomogeneous continuous foundation is studied. The inhomogeneity of the foundation is caused by a slight periodical variation of the foundation stiffness. The moving mass and the beam are assumed to be always in contact. With the help of a perturbation analysis it is shown analytically that vibrations of the system may become unstable. The physical phenomenon that lies behind this instability is parametric resonance that occurs because of the periodic (in time) variation of the foundation stiffness under the moving mass. The first instability zone is found in the system parameters within the first approximation of the perturbation theory. The location of the zone is strongly dependent on the spatial period of the inhomogeneity and on the weight of the moving mass. The larger this period is and/or the smaller the mass, the higher the velocity is at which the instability occurs.

An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice

This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.

Instability of vibrations of an oscillator moving along a beam on an elastic half-space

Abstract The stability of vertical vibrations of an oscillator moving uniformly along a beam on an elastic half-space has been investigated. Expressions for the equivalent stiffness of the beam have been derived at the contact point with the oscillator. It has been shown that the imaginary part of the equivalent stiffness can have a sign, which is interpreted as the so-called ‘negative viscosity’. Frequency bands where the equivalent stiffness gives the ‘negative viscosity’ have been analyzed. Using the expressions for equivalent stiffness the instability zones for the oscillator vibrations have been found. Instability can take place when the velocity of the oscillator exceeds the minimum phase velocity of waves in the beam. The effect of viscosity in the beam on the stability of the system has been considered. It has been shown that a small viscosity destabilizes the system.