Linear Viscoelastic Foundation Research Articles

Hydroelastic analysis of surface gravity wave interacting with elastic plate resting on a linear viscoelastic foundation

This article investigates the interaction between a surface gravity wave that propagates over an elastic plate based on linear viscoelastic foundation. The plate is considered to be thin and infinite and is modeled based on the Euler–Bernoulli beam theory. Static and dynamic boundary conditions are applied to the Laplace equation of the fluid domain. The dispersion relation of the wave–plate system is derived and ratio of surface wave amplitude and plate deflection is proposed. Considering dimensionless dispersion relation, two modes of propagating wave are attained. Problem is analyzed for two cases of presence and absence of viscous damping coefficient in the foundation of the elastic plate. It is shown that flexural rigidity of the submerged plate has considerable effect on wave decay and plate vibration. It is illustrated that shallowness has noticeable effect on the wave propagation frequency and a critical shallowness demarcates damped or overdamped excitation of the elastic plate based on the viscoelastic foundation. Moreover, effects of flexural rigidity of the plate, foundation stiffness coefficient, and foundation viscous coefficient on phase and group velocities of wave are discussed in the present study.

On stability of a slightly curved Maxwell viscoelastic pipe conveying fluid resting on linear viscoelastic foundation

ABSTRACT Viscoelastic pipes are commonly found in various industry applications. When undergoing deformations, a viscoelastic material combines both viscous and elastic behaviours, by exhibiting time-dependent strains. Hamilton’s principle is employed to derive the governing equation for the slightly curved viscoelastic pipe resting on an elastic medium. The pipe’s viscoelasticity is assumed using the Maxwell model. Three trigonometric function are used to describe the initial curvatures, while Eigenfunction expansion method was used to convert the partial differential equation to a set of four ODE’s in time. The Eigenvalues, corresponding to the natural frequencies, were obtained up to four modes. Numerical results show the influence of the initial curvature, viscoelastic coefficient and the linear elastic medium on the frequency and fluid velocity of the pipe. It is noticeable that the initial curvature and the elastic medium cause the bifurcations point of the slightly curved pipe to be extended. For low viscoelasticity coefficient, the results obtained show two dissimilar imaginary frequencies. This is unlike what is previously reported in the literature. For higher coefficient, the pipe behaves like a steel pipe. At higher velocities, coupled mode flutter are absent for some modes depending on the boundary condition.

Modeling and dynamic analysis of a vehicle-flexible pavement coupled system subjected to road surface excitation

Increased road traffic combined with heavy vehicle loads leads to deterioration of pavements and reduces the life span of the paved roads. As a result, large amounts of financial resources are spent each year to improve and maintain road infrastructure around the world. Vehicle dynamics and pavement dynamics are strongly coupled through their contact points. This research focuses on the dynamic analysis of pavement-vehicle interaction system and the effect of coupling action on the response. The system response due to the moving vehicular load on rough road supported by a linear visco-elastic foundation was investigated. The vehicle is modeled as a two-degree-of-freedom quarter-vehicle model, and the pavement-foundation system is described by a simply supported Euler-Bernoulli beam resting on Pasternak foundation, while the tire is coupled to the flexible pavement with a single point contact. Galerkin method was used to develop a system of governing differential equations for a coupled system in the time domain. Direct numerical integration method using Newmark-β based on linear average acceleration method was then used to solve the governing equations and evaluate the response of the coupled system. The results were validated with previous research works and compared with conventional uncoupled systems. Finally, the effects of parameters such as vehicle speed, road roughness, soil stiffness and suspension damping on the responses were investigated.

Dynamic response of timoshenko beam resting on non – linear viscoelastic foundation carrying any number of spring - mass systems

The vibration characteristic of a Timoshenko beam resting on non linear viscoelastic foundation subjected to any number of springs ndash mass systems sprung masses is governed by system of non ndash linear partial differential equations The governing differential equations are examined using differential quadrature method to be transformed with boundary conditions into a set of algebraic equations The non ndash linear Pasternak foundation is assumed to be cubic Therefore the effects of shear deformable beam and the shear deformation of foundations are considered at the same time The numerical investigations show the dynamic response considering different values for engineering properties for both beam and foundation Also the numerical investigations show the efficiency and reliability of using differential quadrature method

Creep and relaxation Poisson’s ratio: Back to the foundations of linear viscoelasticity. Application to concrete

Various scientific communities, mainly in concrete and polymer materials fields, have extended the definition of the Poisson’s ratio to linear viscoelasticity. Depending on the authors, the viscoelastic Poisson’s ratios can be increasing, decreasing or non-monotonic functions of time. Going back to the classic integral formulation of the linear viscoelastic behaviour, creep and relaxation Poisson’s ratios are rederived as functions of bulk and shear relaxation or compliance functions. Both non ageing and ageing behaviours are considered. A literature survey on the thermodynamic restrictions on the viscoelastic characteristics shows that the ageing case has been much less studied than the non ageing case. Still, some examples, both theoretical, including in the ageing case, and practical, regarding concrete, are provided to highlight that any evolution of the viscoelastic Poisson’s ratios is possible: increasing, decreasing and even non monotonous.

Dynamics of Functionally Graded Beams on Viscoelastic Foundation

In the present contribution, the nonlinear dynamics of beams under axial time-dependent excitation is studied. The beams, which rest on a linear viscoelastic foundation, are assumed as axially graded both in terms of geometrical and material properties. The transversal motion equation is derived under two main hypotheses, namely the beams satisfy the requirements of Euler–Bernoulli theory and undergo moderately large deflections (MLDs).

Stability of non-trivial equilibrium paths of beams on a partially visco-elastic foundation

We consider a cantilever beam partially resting on a linear visco-elastic foundation of generalized Winkler type. The length and placement of the partial foundation are variable. The beam is subjected to a sub-tangential force at its unconstrained end. The stability of some of its non-trivial equilibrium configurations is investigated by a numerical procedure based on a finite differences technique. The critical boundaries of buckling and flutter are found; it turns out that the critical conditions for both static and dynamic instability depend on some physical parameters, and interactions between the boundaries of the domains of stability appear.

Effects of parameters on dynamic responses for a heavy vehicle-pavement-foundation coupled system

A three-dimensional heavy vehicle-pavement-foundation coupled system is modelled as a seven DOF vehicle moving along a simply supported double-layer rectangular thin plate on a linear viscoelastic foundation. Using the Galerkin method and a quick direct integral method, the dynamic responses of the coupled system with stochastic road surface roughness are obtained numerically. The effects of vehicle and road parameters on vehicle riding comfort and road damage are also investigated and some interesting conclusions are found which are beneficial to parameter design. In addition, the validity of simulation results is verified by a field test.

Investigation on dynamical interaction between a heavy vehicle and road pavement

This paper presents a model for three-dimensional, heavy vehicle-pavement-foundation coupled system, which is modelled as a seven-DOF vehicle moving along a simply supported double-layer rectangular thin plate on a linear viscoelastic foundation. The vertical tyre force is described by a single point-contact model, while the pavement-foundation is modelled as a double-layer plate on a linear viscoelastic foundation. Using the Galerkin method and quick direct integral method, the dynamical behaviour of the vehicle-pavement-foundation coupled system is investigated numerically and compared with that of traditional vehicle system and pavement system. The effects of coupling action on vehicle body vertical acceleration, suspension deformations, tyre forces and pavement displacements are also obtained. The investigation shows that the coupling action could not be neglected even on a smooth road surface, such as highway. Thus, it is necessary to investigate the dynamics of vehicle and pavement simultaneously based on the vehicle-pavement-foundation coupled system.

Dynamics of vehicle-pavement coupled system based on a revised flexible roller contact tire model

A revised flexible roller contact tire model (RFRC tire model) is proposed, which considers not only the geometric and flexible filtering effect, but also tire damping and pavement displacement. A vehicle-pavement coupled system is modeled as a two DOF oscillator moving along a simply supported beam on a linear viscoelastic foundation. By using the Galerkin’s and Direct Integral method, dynamical responses of the vehicle-pavement coupled system are obtained based on the RFRC tire model and the traditional single point contact tire model (SPC tire model). The simulation results are compared with test data and the validity of the proposed RFRC tire model is verified. Differences between the two models are also investigated. It is found that the dynamical behaviors for both models agree with each other quite well when road surface roughness is a long harmonic wave. On the other hand, they are different under short harmonic wave or impulse road excitation. Thus the RFRC tire model should be used to compute the tire force and investigate dynamical responses of vehicle and pavement.

Dynamic Analysis of Concrete Pavements Subjected to Moving Loads

Dynamic response of concrete pavements subjected to moving loads is analyzed by using the three-dimensional (3D) finite-element method in conjunction with Newmark integration scheme. The dynamic vehicle-pavement-foundation interaction effects are considered in the 3D finite-element algorithm. The moving vehicle loads are modeled as lumped masses each supported by a spring-dashpot suspension system and having a specified horizontal velocity and acceleration. Concrete pavements are considered to respond elastically and are represented by a series of brick elements. The present formulation considers a linear viscoelastic foundation model (Kelvin model) consisting of a system of discrete linear springs and dashpots. The interaction between concrete pavements and underlying soil foundation is considered. The present results are compared with the available theoretical results to validate the accuracy of the present algorithm. A study is conducted to evaluate the effects of various parameters on the dynamic response of concrete pavements subjected to moving loads.

Remarks on the foundations of linear viscoelasticity

Following Coleman and Noll, the relationship between stress and the history of strain in linear visco-elasticity is established as a linearized version of that for a simple material with fading memory. However, the necessary introduction of a Hilbert space structure on the set of histories requires the extension of the domain of a response function to a space of histories in which non-physical histories predominate. Consequently, no natural extension exists and, a priori, the linearization may be an artefact of the extension procedure. Here both linearized stress values and the stress relaxation function are shown to depend only upon the original response function.

Bending of a circular plate on a linear viscoelastic foundation

Bending of a circular plate on a linear viscoelastic foundation

Nonlinear Viscoelastic Solids

1. Introduction. The classical theories of the mechanics of deformable materials are, for the most part, linear theories and ones in which the stress depends only on the kinematic situation existing at the time of measurement of the stress. For many materials, such as viscoelastic solids and liquids, these assumptions are not valid. Since the paper by Oldroyd [13], considerable advances have been made in formulating the continuum mechanics of such materials. In the present paper an account of this work will be given, particularly in relation to viscoelastic solids. Provided that any characteristic lengths which may exist iu the solid can be ignored, we may take as our starting point in the development of appropriate constitutive equations the assumption that the stress tensor: depends not only on the displacement gradients existing in the material considered at the instant at which the stress is measured, but also on those which existed in it at all previous times, i.e., the stress is a functional of the deformation gradient history up to the instant of measurement. If some reference state is adequately specified, the assumption that stress depends on the history of the deformation gradients and the assumption that stress depends on the history of the velocity or acceleration. gradients are quite equivalent, since either of the latter determine the deformation gradient history. The form of the functional dependence of the stress on the deformation gradient history cannot be arbitrarily chosen. It is subject to two major restrictions. The first of these, which is discussed in 2, arises from the fact that a simultaneous time-dependent rotation of the deformed body and reference system leaves the stress components unaltered. The second, which is discussed in 3 and 5, arises from any symmetry which the material may possess in its reference state. The important particular case when the material is isotropic in its reference state is discussed in 5. In certain circumstances, the history of the deformation gradients may be adequately described by a knowledge of the values of the deformation gradients and of the gradients of velocity, acceleration, second acceleration., etc., at the instant of measurement of the stress. The development of canonical forms for the constitutive equations in this case is discussed generally in 4 and for the particular circumstance when the material is isotropic in its reference state in 5. The constitutive equations which arise from these general studies are inevitably complicated. In 6 attention is drawn to the great simplifications whichmay arise

Foundations of Linear Viscoelasticity

Foundations of Linear Viscoelasticity

The Infinite Elastic Beam on a Linear Viscoelastic Foundation

The infinite, elastic beam on three different types of linear viscoelastic foundations is analyzed under various loads. Initial, transient, and terminal states of beam deflection and foundation pressure are determined; conditions on the load function are stated. Numerical examples with graphs are given. The extension of the results to finite beams is discussed.