Viscoelastic Foundation Model Research Articles

Stress analysis of adhesively-bonded single stepped-lap joints based on three-parameter fractional viscoelastic foundation model

This paper aims to develop a viscoelastic analytical model for adhesively bonded single stepped-lap joints subjected to tensile loading. The adherends are aluminum alloy A6063 and modeled as Timoshenko elastic beams and the adhesive is epoxy type B. A three-parameter fractional viscoelastic foundation (3PFVF) model is proposed to express the governing stresses in the joint and the fractional Zener model is used to model the viscoelastic behavior of the adhesive layer. The proposed 3PFVF model makes it possible to have different peel stresses between the two interfaces of adhesive and adherends. The governing differential equations are derived in the Laplace domain, and then solved and transformed simultaneously in the time domain using the Gaver-Stehfest inverse Laplace transform method. The finite element simulation with ANSYS is applied to validate the proposed method. The results show that a simple fractional viscoelastic model, which has a short differential equation, offers the same results as the classical viscoelastic models, which have higher and more complex differential equations. Moreover, the results show that the maximum shear and peel stresses in the single stepped-lap joints are about 20% less than single-lap joints.

Deformation response of roof in solid backfilling coal mining based on viscoelastic properties of waste gangue

Solid backfill mining (SBM) is a form of green mining, the core of which is to control and minimize the deformation and movement of strata above longwall coal mines. Establishing a mechanical model that can reliably describe roof deformation by considering the viscoelastic properties of waste gangue is important as it assists in improving mine designs and reducing the environmental impact on the surface. In this paper, the time-dependent deformation characteristics of gangue under different stress levels were obtained by using lateral confinement compression, that reliably represents the compaction of goaf. The viscoelastic foundation model for gangue mechanical response is different from the traditionally used elastic foundation model, as it considers the time factor and viscoelasticity. A mechanical model using a thin plate on a fractional viscoelastic foundation was established, and the roof deflection, bending moment, time-dependent, viscous and other characteristics of SBM were included and analyzed. Compared with the existing elastic foundation model, the proposed fractional order viscoelastic foundation model has higher accuracy with laboratory data. The plate deflection increases by 50.9% and the bending moment increases by 37.9% after 100 days, which the elastic model would not have been able to predict.

The Scheme to Determine the Convergence Term of the Galerkin Method for Dynamic Analysis of Sandwich Plates on Nonlinear Foundations

The vibration of a plate resting on elastic foundations under a moving load is of great significance in the design of many engineering fields, such as the vehicle-pavement system and the aircraft-runway system. Pavements or runways are always laminated structures. The Galerkin truncation method is widely used in the research of vibration. The number of truncation terms directly affects the convergence and accuracy of the response results. However, the selection of the number of truncation terms has not been clearly stated. A nonlinear viscoelastic foundation model under a moving load is established. Based on the natural frequency of linear undisturbed derivative systems, the truncation terms are used to determine the convergence of vibration response. The criterion for the convergence of the Galerkin truncation term is presented. The scheme is related to the natural frequency with high efficiency and practicability. Through the dynamic response of the sandwich beam under a moving load, the feasibility of the scheme is verified. The effects of different system parameters on the scheme and the truncation convergence of dynamic response are presented. The research in this paper can be used as a reference for the study of the vibration of elastic foundation plates. Especially, the model established and the truncation analysis method proposed are helpful for studying the vibration of vehicle-pavement system and related systems.

Vibration Analysis of Viscoelastic FGM Nanoscale Plate Resting on Viscoelastic Medium Using Higher-order Theory

The present article aims essentially to present an analytical and numerical method which makes it possible to study the damped vibrations of viscoelastic FGM nanoplates resting on viscoelastic foundations. A new model for the higher-order shear deformation plate theory is coupled with the internal Kelvin - Voigt viscoelastic model and the three-parameter viscoelastic foundation model for the purpose of reducing and minimizing the vibration response of the system. It is widely admitted that the mechanical properties of these new functionally gradient materials (FGMs) vary according to the thickness of the plate and depend on its volume fraction. The use of FGM plates seems to be an ideal solution for the study of free vibrations because of their multifunctionality that is fully integrated with the nonlocal Eringen effect. The dynamic response of such a complex system has been investigated by varying the aspect ratio of the plate, the mechanical characteristics of the material used, the internal and external damping and the foundation rigidity. The results obtained, with and without the nonlocal effect, were compared with those of different models of higher-order theories and under various boundary conditions; they were found to be in good agreement with those reported in the literature.

Dynamic responses of long tunnels in layered viscoelastic ground subjected to inclined SH waves

Dynamic responses of a long tunnel in a horizontally layered viscoelastic ground subjected to inclined SH waves are analyzed by Fourier transform in this paper. The tunnel is assumed to be a beam. The free field displacement of the horizontally layered viscoelastic ground is imposed on the long tunnel through the viscoelastic foundation model. The analytical solution is verified through comparison with numerical solutions and other solutions in the literature. Based on the solution, the difference between rigid bases and elastic bases is clarified, and the influence of pressure-dependent effect is discussed. Parametric analyses on the incident angle and the flexibility ratio are also performed. The solution can be easily used in the preliminary seismic analysis and design of tunnels.

Three-parameter viscoelastic foundation model of adhesively bonded single-lap joints with functionally graded adherends

Functionally graded (FG) adherends are beneficial to adhesively bonded joints, as different material composition through the adherend thickness can manipulate the interfacial stress distribution. It is a fact that both shear and peeling stresses along the adhesive/adherend interface at the free edges play important roles in the structural integrity of a joint. Previous studies have shown that both shear and peeling stresses can be uniformly distributed near the edges of a joint with FG adherends if the right adherend composition is selected through the adherends’ thickness. However, the effect of the viscoelastic behavior of the adhesive layer has been neglected in those studies. This study establishes a viscoelastic analytical model for adhesively bonded single-lap joints with FG adherends. In this model, the adhesive layer is simulated as a three-parameter viscoelastic foundation using a standard linear solid (SLS) model to account for both creep and relaxation behaviors in the adhesive. This model satisfies the zero-shear-stress boundary at the free edges of the adhesive and predicts different peel stresses along two adherend/adhesive interfaces. Excellent agreement with finite element analysis (FEA) has been achieved by the present model, confirming the accuracy of the model. The viscoelastic behavior of the adhesive layer affects stress concentration near the edges of a joint at early ages, suggesting that the present model can properly capture the stress relaxation in adhesively bonded joints with FG adherends. The parametric studies show that FG material configuration and the mechanical properties of adhesive layers play an important role in the uniformity of shear stress distribution along the length of single-lap joints with FG adherends. The present viscoelastic solution can predict more uniform stress distribution and is a valuable tool in design optimization of joints with FG adherends.

Nonlocal elasticity and shear deformation effects on thermal buckling of a CNT embedded in a viscoelastic medium

The thermal buckling analysis of carbon nanotubes embedded in a visco-Pasternak’s medium is investigated. The Eringen’s nonlocal elasticity theory, in conjunction with the first-order Donnell’s shell theory, is used for this purpose. The surrounding medium is considered as a three-parameter viscoelastic foundation model, Winkler-Pasternak’s model as well as a viscous damping coefficient. The governing equilibrium equations are obtained and solved for carbon nanotubes subjected to different thermal and mechanical loads. The effects of nonlocal parameter, radius and length of nanotube, and the three foundation parameters on the thermal buckling of the nanotube are studied. Sample critical buckling loads are reported and graphically illustrated to check the validity of the present results and to present benchmarks for future comparisons.

Transverse Vibration of Tapered Single-Walled Carbon Nanotubes Embedded in Viscoelastic Medium

Based on the nonlocal theory, Euler-Bernoulli beam theory and Kelvin viscoelastic foundation model, free transverse vibration is studied for a tapered viscoelastic single-walled carbon nanotube (visco-SWCNT) embedded in a viscoelastic medium. Firstly, the governing equations for vibration analysis are established. And then, we derive the natural frequencies in closed form for SWCNTs with arbitrary boundary conditions by applying transfer function method and perturbation method. Numerical results are also presented to discuss the effects of nonlocal parameter, relaxation time and taper parameter of SWCNTs, and material property parameters of the medium. This study demonstrates that the proposed model is available for vibration analysis of the tapered SWCNTs-viscoelastic medium coupling system.

Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium

The effect of longitudinal magnetic field on vibration response of a sing-walled carbon nanotube (SWCNT) embedded in viscoelastic medium is investigated. Based on nonlocal Euler-Bernoulli beam theory, Maxwell’s relations, and Kelvin viscoelastic foundation model, the governing equations of motion for vibration analysis are established. The complex natural frequencies and corresponding mode shapes in closed form for the embedded SWCNT with arbitrary boundary conditions are obtained using transfer function method (TFM). The new analytical expressions for the complex natural frequencies are also derived for certain typical boundary conditions and Kelvin-Voigt model. Numerical results from the model are presented to show the effects of nonlocal parameter, viscoelastic parameter, boundary conditions, aspect ratio, and strength of the magnetic field on vibration characteristics for the embedded SWCNT in longitudinal magnetic field. The results demonstrate the efficiency of the proposed methods for vibration analysis of embedded SWCNTs under magnetic field.

Vibration analysis of viscoelastic single-walled carbon nanotubes resting on a viscoelastic foundation

Vibration responses were investigated for a viscoelastic Single-walled carbon nanotube (visco-SWCNT) resting on a viscoelastic foundation. Based on the nonlocal Euler-Bernoulli beam model, velocity-dependent external damping and Kelvin viscoelastic foundation model, the governing equations were derived. The Transfer function method (TFM) was then used to compute the natural frequencies for general boundary conditions and foundations. In particular, the exact analytical expressions of both complex natural frequencies and critical viscoelastic parameters were obtained for the Kelvin-Voigt visco-SWCNTs with full foundations and certain boundary conditions, and several physically intuitive special cases were discussed. Substantial nonlocal effects, the influence of geometric and physical parameters of the SWCNT and the viscoelastic foundation were observed for the natural frequencies of the supported SWCNTs. The study demonstrates the efficiency and robustness of the developed model for the vibration of the visco-SWCNT-viscoelastic foundation coupling system.

Free transverse vibration of double-walled carbon nanotubes embedded in viscoelastic medium

This paper aims to study the vibration characteristics of viscoelastic double-walled carbon nanotubes embedded in a viscoelastic medium. In doing this, the governing equations of the system are derived by combining the Euler–Bernoulli beam theory, nonlocal viscoelastic model and Kelvin viscoelastic foundation model. Subsequently, the transfer function method is employed to solve the governing equations, which enables one to obtain the natural frequencies and the corresponding mode shapes in closed form for the DWCNTs with arbitrary boundary conditions. Here, the developed mechanics model is first compared with the existing techniques available in the literature, where excellent agreement is achieved. Also, a detailed parametric study is conducted to examine the effect of boundary conditions, nonlocal parameter, relaxation time, slenderness ratio, stiffness coefficient and damping coefficient on the vibration response of the DWCNTs.

Analysis of change in motional capacitance of electrical equivalent circuit of quartz-crystal tuning-fork tactile sensor induced by viscoelastic materials in contact with its base

We investigated experimentally and theoretically the change in the reciprocal of the motional capacitance Δ(1/Ca) of a quartz-crystal tuning-fork tactile sensor before and after its base comes into contact with neoprene rubbers. We derived the analytical formula for the motional capacitance of the electrical equivalent circuit of the sensor at resonance based on the combination of the L-shaped bar model and viscoelastic foundation model, and the law energy of conservation. We found from both contact experiments and theoretical considerations on the analytical formula that Δ(1/Ca) is intrinsically induced by both the dynamic Young’s modulus and viscosity of neoprene rubbers at 32.5 kHz, which is the actual resonant frequency of the quartz tactile sensor. In this paper, we showed experimentally and theoretically for the first time that the motional capacitance of the tuning fork tactile sensor is affected by the dynamic viscosity of materials in contact with the sensor’s base, except their dynamic Young’s modulus.

Interface stress redistribution in FRP-strengthened reinforced concrete beams using a three-parameter viscoelastic foundation model

External bonding of FRP plates or sheets has become a popular method for strengthening reinforced concrete structures. Stresses along the FRP–concrete interface are critical to the effectiveness of this technique because high stress concentration along the FRP–concrete interface can lead to the FRP debonding from the concrete beam. In this study, a novel analytical solution has been developed to predict the interface stress redistribution of FRP-strengthened reinforced concrete beams induced by the viscoelastic adhesive layer. Both the FRP plate and the RC beam are modeled as Timoshenko’s beams, connected through the adhesive layer. The adhesive layer is modeled as a three-parameter viscoelastic foundation (3PVF) using Standard Linear Solid model. The 3PVF model satisfies the equilibrium equation of the adhesive layer and the zero shear-stress boundary condition at the free edge. Closed-form expressions of the time-dependent interface stresses and deflection of the beam are obtained using Laplace transform. Finite element analysis is also conducted to verify the analytical solution using a subroutine UMAT based on the Standard Linear Solid model. Numerical results suggest that the stress concentrations within the FRP–concrete interface relax with time. The axial force in the FRP plate also reduces with time due to the creep of the adhesive layer. However, this relaxation is limited to a small zone close to the end of the FPR plate.

Axial Vibration Analysis of Buried Pipeline

The pipe model is simplified as elastic foundation beam model of Euler-Bernoulli in the paper. The supported forms are fixed support and free support in the analysis of axial vibration. Kelvin viscoelastic foundation model is adopted and the dynamic model of soil spring is regarded as linearity. Applying the principle of Hamilton, the differential equation of axial vibration is deduced. By utilization of the first three-order modal and the orthogoality of main vibration mode, the equations of earthquake excitaiton are discreted and transformed into common form of dynamic equation. A typical numerical example is analyzed by using of the Matlab software.

Establishment of Mathematical Model of Buried Pipeline on Nonlinear Soil Dynamic Model

The pipe model is simplified as elastic foundation beam model of Euler-Bernoulli in the paper, the supported form of pin rocker bearing in the analysis of transverse vibration. Kelvin viscoelastic foundation model is adopted and the dynamic model of soil spring is regarded as nonlinearity. Applying the principle of Hamilton, the differential equation of vibration is deduced. By utilization of the first three-order modal and the orthogoality of main vibration mode, the equations are discreted and transformed into state formulas. The critical flow velocity is obtained using the Matlab software in a typical numerical example.

Free Vibration of Elastic Timoshenko Beam on Fractional Derivative Winkler Viscoelastic Foundation

The fractional derivative Winkler viscoelastic foundation model is established by introducing the concept of fractional derivative. The control equations of free vibration of elastic Timoshenko beam on fractional derivative Winkler viscoelastic foundation are also built by considering the shear deformation and rotary inertia, and the control equations of elastic Timoshenko beam are decoupled by using the deformation function and considering the properties of fractional derivative, and the expressions of deflection and section corner of elastic Timoshenko beam on fractional derivative Winkler viscoelastic foundation are obtained. The influences of fractional derivative order and shear shape factor on the free vibration of elastic Timoshenko beam are discussed by numerical example.

Vibration analysis of beams with non‐local foundations using the finite element method

AbstractIn this paper, a non‐local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the non‐local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two‐node beam elements. However, for non‐local elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiffness matrices. The expressions of these direct and cross‐matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions. Alternatively numerical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler–Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions and Galerkin approximations. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non‐local viscoelastic foundations. Copyright © 2007 John Wiley & Sons, Ltd.

The effect of tangential elasticity of the contact layer between stator and rotor in travelling wave ultrasonic motors

In travelling wave ultrasonic motors the elliptical motion of material points of the stator drives the rotor due to frictional mechanisms. The motor characteristic strongly depends on the mechanical properties of the components stator, rotor and contact layer. In order to predict the motor behaviour, a model for the contact between stator and rotor has been developed. The goal of the present paper is to point out the importance of the tangential elasticity of the contact layer which is responsible for the formation of stick zones and also for the amount of friction losses and overall efficiency. Therefore a comparison with a model with a contact layer rigid in tangential direction is given. Based on a visco-elastic foundation model for the contact layer, torque–speed curves as well as torque–efficiency curves are computed. Experimental investigations for identification of parameters, check of assumptions and model validation are carried out. Finally, the model is used to show the results of parameter variations for normal force, vibration amplitude and modulus of elasticity of the contact layer.

Receptance behaviour of railway track and subgrade

Vertical dynamic behaviour of a railway track on an elastic halfspace or on a layered halfspace is investigated by a frequency domain analysis. The results are compared with those for a simpler model, where ballast and subgrade are considered as a viscoelastic foundation. In the low- and medium-frequency range up to 250 Hz, great differences are observed between the results of the halfspace model and the results of the viscoelastic foundation model. This is because the damping due to wave propagation and coupling between sleepers cannot be modelled correctly by a viscoelastic foundation. Contradictions observed in the past between measured and calculated results can be explained with the new halfspace model. For frequencies higher than 250 Hz, the influence of the subgrade is negligible, so that here the simpler viscoelastic foundation model can be used.

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.