Viscoelastic Foundation Research Articles

Thermoelastic Response of Microbeams under a Magnetic Field Rested on Two-Parameter Viscoelastic Foundation

Mechanical foundations and pavements design need a perfect prediction of the response of the material to reach a reliable and safe structure. This work deals with the thermoelastic response of microbeams rested on a two-parameter viscoelastic foundation due to a magnetic field in the context of the dual - phase lag thermoelasticity model. The solutions of the governing equations are attained using the Laplace transform method. The distributions of the deflection, temperature, the displacement and the flexure moment of the micro beam are numerically obtained and illustrated graphically. The effects of the magnetic field, Winkler and shear foundation parameters, the ramping time parameter and the models of thermoelasticity on the considered fields are concerned and discussed in details. For comparison purposes, the response of the micro beam and the dynamic deflection using the Bernoulli- Euler beam and thermoelasticity theories are compared with earlier investigated studies and magnificent agreements are detected.

Dynamic stability of nonlocal strain gradient FGM truncated conical microshells integrated with magnetostrictive facesheets resting on a nonlinear viscoelastic foundation

The objective of this investigation is to predict the size-dependent dynamic stability of truncated conical microshells made of a functionally graded material (FGM) integrated with magnetostrictive facesheets. The microshells are subjected to a combination of axial compressive load and magnetic field in the presence of the nonlocality and strain gradient size dependencies. The conical microshells are assumed to be surrounded by a two-parameter Winkler-Pasternak medium augmented via a Kelvin-Voigt viscoelastic approach taking a nonlinear cubic stiffness into account. The nonlocal strain gradient-based differential equations of motion are constructed based upon the third-order shear deformation conical shell theory including the magnetic permeability tensor together with the magnetic fluxes. The discretization process within the framework of the generalized differential quadrature technique is employed to achieve the nonlocal strain gradient-based load-frequency responses. It is found that increasing the material gradient index results in to decrease the nonlocal strain gradient frequency obtained for a specific value of the axial compression within the prebuckling regime. However, within the postbuckling domain, an increment in the value of the material gradient index plays an opposite role. Also, the gap between the load-frequency curves associated with various material gradient indexes are more prominent for the nonlocal strain gradient cases in comparison with the classical one.

Nonlinear Creep Model of Deep Gangue Backfilling Material and Time-Dependent Characteristics of Roof Deformation in Backfilling Mining

With the gradual increase in mining depth of coal resource exploitation, deep backfilling mining has effectively solved the impact of strong deep mine pressure and strong mining disturbances. However, after deep backfilling mining, the backfilling material is subjected to high stress for a long time, and its viscoelasticity has a significant impact on the roof control effect. This paper uses a large-scale bulk confinement test device to analyze the viscoelastic properties of gangue, establishes a high-precision fractional viscoelastic creep model, and identifies the gangue parameters. The established fractional viscoelastic model was used as the foundation model of the beam, and the roof model based on the fractional viscoelastic foundation was solved. The top deformation characteristics of elastic foundation and fractional foundation were compared and analyzed, and the time effect, viscoelastic effect, and order effect of the fractional order viscoelastic foundation beam were discussed. The results show that the viscosity of gangue increased under the action of deep high stress. As time increased, the roof deformation also increased. In order to more effectively control the long-term deformation of the roof, the viscosity coefficient of the backfilling material should be greater than 20 MPa. This research provides relevant guidance for the requirements of backfilling materials for deep backfilling mining and the prediction of long-term dynamic deformation of the roof in underground excavations.

A New Beam Element for Static, Free and Forced Vibration Responses of Microbeams Resting on Viscoelastic Foundation Based on Modified Couple Stress and Third-Order Beam Theories

In this paper, a new beam element based on third-order Reddy beam theory and modified couple stress theory is introduced for static and dynamic analyses of microbeams. The beam is resting on Kelvin–Voigt-type viscoelastic foundation. Equilibrium equations are derived by using Hamilton’s principle. Galerkin finite element method is applied to the governing equations, and shape functions of the new beam element have been derived. As a result, applying only six elements are suitable for assessing beam behavior precisely, and it is beneficial due to the reduction in computation time. The proposed element is a two-node element with 4 degrees of freedom at each node. The effect of different stiffness and damping of viscoelastic foundation, boundary condition and length parameters on natural frequency and transient response of the microbeam is investigated.

Vibration control of a smart shell reinforced by graphene nanoplatelets under external load: Semi-numerical and finite element modeling

In this article, smart control and frequency characteristics of a graphene nanoplatelets composite (GPLRC) cylindrical shell surrounded by piezoelectric layers as sensor and actuator (PLSA) are presented. The current structure is under an external load. For the semi-numerical method, the strain-stress relations can be determined through the first-order shear deformable theory (FSDT). For access to various mass densities as well as the Poisson ratio, the rule of the mixture is applied, although the modified Halpin-Tsai theory for obtaining the module of elasticity. The external voltage is applied to the sensor layer, while a Proportional-Derivative (PD) controller has been utilized for controlling the output of the sensor. The boundary conditions are derived through governing equations of the GPLRC cylindrical shell surrounded by PLSA using an energy method known as Hamilton's principle and finally are solved using a generalized differential quadrature method (GDQM). Apart from a semi-numerical solution, a finite element model was presented using the finite element package to simulate the response of the smart GPLRC cylindrical shell. The results created from a finite element simulation illustrates a close agreement with the semi-numerical method results. The outcomes show that the PD controller, viscoelastic foundation, slenderness factor (L/R), external voltage, and GPL's weight fraction have a considerable impact on the amplitude and vibration behavior of a GPLRC smart cylindrical shell. As an applicable result in related industries, the parameter and consideration of the PD controller have a positive effect on the static and dynamic behaviors of the structure subjected to an external load.

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.

Dynamics of a Euler–Bernoulli beam on nonlinear viscoelastic foundations: a parameter space analysis

The dynamics and spectrum of vibrations of an uniform Euler–Bernoulli beam on nonlinear viscoelastic foundations are investigated in parameter space. An alternative method of approach based on mean-field approximation is proposed. Spatial and temporal solution are obtained in an interactive way. The spectrum is reached directly from spatial solutions. It was shown that the system exhibits sudden birth and death of chaotic attractors (crises) and that there are periodic windows immersed in chaotic regions. It was determined the regions in the parameter space where chaotic and regular attractors occur. Moreover, the model presents the coexistence of many attractors. The numerical simulations found geometrical structures known as shrimps, and these periodic structures are involved by chaotic regions. In this work it was investigated the dynamics of a beam of Euler–Bernoulli with viscous and dynamic support. It is presented, then, a model to treat this type of problem, which is nonlinear and non-autonomous. The dynamics of the model presents a complex behavior, being in some cases sensitive to the initial conditions. Changing properly the parameters the dynamics goes from chaotic behavior to periodic regime. As the system is forced and dissipative, complex structures in the parameter space, known as shrimps, arise. Also, it was found that the spatial solution is not sensitive to the dynamics, although they are coupled. However, the dynamics is very sensitive to the mean quadratic deformation of the beam and, therefore, it is related to the vibration modes obtained in the spatial solution.

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.

Static and dynamic hygrothermal postbuckling analysis of sandwich cylindrical panels with an FG-CNTRC core surrounded by nonlinear viscoelastic foundations

In this article, analytical and semi-analytical methods are used to analyze the nonlinear static and dynamic hygrothermal postbuckling response of sandwich cylindrical panels with functionally graded carbon nanotube-reinforced composite (FG-CNTRC) core. The sandwich cylindrical panels with an FG-CNTRC core are resting on generalized nonlinear viscoelastic foundations which are consisted of a Winkler and Pasternak foundation parameters augmented by a Kelvin-Voigt viscoelastic model and a nonlinear cubic stiffness. The sandwich cylindrical panels with four types of FG-CNTRC core are considered such that the CNTs distribution types consist of the FG-O, FG-Λ, FG-X, and UD. Regarding to the classical shell theory and geometrical nonlinearity in von Kármán-Donnell sense, the governing equation is obtained and discretized utilizing the Galerkin method. The nonlinear dynamic hygrothermal (NDHT) postbuckling is analyzed by means of the fourth-order Runge-Kutta method. The influences of material parameters, various geometrical characteristics, and nonlinear viscoelastic foundation parameters are studied on the nonlinear static hygrothermal (NSHT) postbuckling and NDHT postbuckling analysis of sandwich cylindrical panels with an FG-CNTRC core. The results show that the different types of CNTs distribution and nonlinear viscoelastic foundation parameters have a strong effect on the hygrothermal postbuckling behaviors of sandwich cylindrical panels with an FG-CNTRC core.

Vibrational characteristics of a higher-order laminated composite viscoelastic annular microplate via modified couple stress theory

This article deals with the vibrational characteristics of a laminated composite annular microplate using a non-classical continuum theory called modified couple stress theory (MCST). The structure is covered with a viscoelastic foundation that is simulated via the Kelvin-Voight model. The non-classical governing equations and boundary conditions of the size-dependent annular microplate are derived by adding the symmetric rotation gradient and higher-order stress tensors to the strain energy. The current non-classical model is capable of capturing the size-dependency in the annular microplate by using only one material length scale parameter; moreover, the mathematical formulation of annular microplate via the classical model can be recovered from the present model by neglecting the material length scale parameter. Finally, the non-classical governing equations are solved using the generalized differential quadrature method (GDQM) for various boundary conditions. Afterward, a parametric study is carried out to investigate the effects of the length scale parameter, radius ratio, circumferential and radial mode number, geometry of the laminated layer, and boundary conditions on the frequency responses of the annular microplate by considering MCST. The results show that in the grater elastic properties ratio (E1/E2) factor, increasing the viscoelastic parameter cannot make any changes in the frequency of the disk.

Influence of Winkler and viscoelastic foundation on free vibration of functionally graded beam integrated with Terfenol-D layer

Free vibration of functionally graded beam integrated with Terfenol-D layer on Winkler-two parameter and viscoelastic foundation are studied by means of differential quadrature method within the framework of classical beam theory. The material properties of functionally graded beam integrated Terfenol-D layer are estimated by using the rule of mixture. The applied kinematic boundary conditions are implemented using $$\delta $$ -point and modified weighting coefficient approach. The solution of simply supported functionally graded material beam with Terfenol-D layer (FGMT) resting on the Winkler elastic foundation is obtained by using the technique of Navier. The numerical results obtained using differential quadrature method (DQM) and modified differential quadrature method (MDQM) are compared with exact results obtained from analytical formulation where excellent agreement is observed. The parametric study is carried out to encapsulate the influence of Winkler-two parameter and the viscoelastic foundation on the vibration characteristics of functionally graded beams integrated with Terfenol-D layer.

Free and Forced Vibration of Coupled Beam Systems Resting on Variable Viscoelastic Foundations

This paper presents a modified variational method for free and forced vibration analysis of coupled beam systems resting on various viscoelastic foundations. Non-uniform as well as uniform curved and straight Timoshenko beam components are considered in the coupled beam system. Using proper coordinate transformations, interactions among the beam components of the coupled beam system are accommodated by combining Lagrange multiplier method and least-square weighted residual method. Interface potential energy for various boundary conditions including the elastic ones is simultaneously formulated. Thus, the proposed method allows flexible choice of the admissible functions, regardless of the boundary conditions. Based on the proposed energy method, Winkler, Pasternak or even variable foundations distributed in a parabolic or sinusoidal manner can be easily introduced into the coupled beam systems. Two kinds of damping, namely the proportional and viscous damping, are also employed to model the energy dissipation of the viscoelastic foundations. Corresponding finite element (FE) simulations are performed where possible and good agreement is observed. Thus, great efficiency and accuracy of the present approach are demonstrated for free, steady-state and transient vibration of the coupled beam systems. The influences of the parameters of the variable viscoelastic foundations on the dynamic properties of the coupled beam system are also examined.

Couple stress-based dynamic stability analysis of functionally graded composite truncated conical microshells with magnetostrictive facesheets embedded within nonlinear viscoelastic foundations

In this paper, size-dependent dynamic stability of axially loaded functionally graded (FG) composite truncated conical microshells with magnetostrictive facesheets surrounded by nonlinear viscoelastic foundations including a two-parameter Winkler–Pasternak medium augmented via a Kelvin–Voigt viscoelastic approach is analyzed considering nonlinear cubic stiffness. To this purpose, von Karman-type kinematic nonlinearity along with modified couple stress theory of elasticity was applied to third-order shear deformation conical shell theory in the presence of magnetic permeability tensor and magnetic fluxes. The numerical technique of generalized differential quadrature (GDQ) was used for the solution of microstructural-dependent dynamic stability responses of FG composite truncated conical microshells. It was seen that moving from prebuckling to postbuckling domain somehow increased the significance of couple stress type of size dependency on frequency. In addition, within both prebuckling and postbuckling regimes, an increase of material gradient index decreased the importance of couple stress type of size dependency on the frequency of an axially loaded FG composite truncated conical microshell. Furthermore, it was revealed that by applying a positive magnetic field to an axially loaded truncated conical microshell with magnetostrictive facesheets, its frequency at a specific axial load value was increased in prebuckling domain and decreased in postbuckling domain. However, this pattern was reversed by applying a negative magnetic field.

Vibration Analysis of Size-Dependent Piezoelectric Nanobeam Under Magneto-Electrical Field

The damping vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam resting on viscoelastic foundation based on nonlocal strain gradient elasticity theory (NSGT) is studied in this article. For this purpose, by considering the effects of Winkler-Pasternak, the viscoelastic medium consists of linear and viscous layers. with respect to the displacement field in accordance with the refined shear deformable beam theory (RSDT) and the Kelvin-Voigt viscoelastic damping model, the governing equations of motion are obtained using Hamilton’s principle based on nonlocal strain gradient theory (NSGT). Using Fourier Series Expansion, The Galerkin’s method adopted to solving differential equations of nanobeam with both of simply supported and clamped boundary conditions. Numerical results are obtained to show the influences of nonlocal parameter, the length scale parameter, slenderness ratio and magneto-electro-mechanical loadings on the vibration behavior of nanobeam for both types of boundary conditions. It is found that by increasing the magnetic potential, the dimensionless frequency can be increased for any value of the damping coefficient and vice versa. Moreover, negative/positive magnetic potential decreases/increases the vibration frequencies of thinner nanobeam. Also, the vibrating frequency decreases and increases with increasing nonlocal parameter and length scale parameter respectively.

Frequency and buckling responses of a high-speed rotating fiber metal laminated cantilevered microdisk

In this article, amplitude, and vibrational characteristics of a rotating fiber metal laminated microdisk are presented. The current microstructure is modeled as a flexible microdisk surrounded by the two-parameter viscoelastic foundation. The centrifugal and coriolis effects due to the rotation are considered. The strains and stresses can be determined via the third-order shear deformable theory. For accessing to size-effects, the nonlocal strain gradient theory is used. The boundary conditions are derived through governing equations of the laminated rotating microdisk using an energy method known as Hamilton’s principle and finally are solved using a numerical method based generalized differential quadrature method.

Nonlinear random vibrations of functionally graded porous nanobeams using equivalent linearization method

In the present study, the nonlinear random vibration of a porous functionally graded nanobeam supported by a viscoelastic foundation has been investigated by making use of the statistical linearization method for the first time. Nonlocal Euler-Bernoulli beam theory with von Kármán type nonlinearity is utilized to derive the governing equation of the motion of the FG porous nanobeam. Two types of porosity distribution have been considered for the FG beam. To discretize the governing equation Galerkin's method is applied. The mean square value of the response of the resulted Duffing's equation in the presence of a random input with nonzero mean value has been extracted using statistical linearization method. The results of the present study are compared with the analytical results of the FPK approach and the results of the perturbation method for the classical limit case of the homogeneous beam. It is revealed through detailed numerical analysis that the statistical linearization method (in contrast to the perturbation method) gives reliable results and there is an excellent agreement between its results and the analytical results. The effect of various parameters on the mean value of the response is also investigated numerically and illustrated graphically. It is found that the mean square value of vibrations increases when porosity, power-law index, nonlocal parameter, or the mean value of the input increases.

The Vibration of Simply Supported Non-Uniform Cross Sectional Pipe Conveying Fluid Resting on Viscoelastic Foundation

The induced flexural vibration of slender pipe systems with continuous non uniform cross sectional area containing laminar flowing fluid lying on extended Winkler viscoelastic foundation is considered. The Euler Bernoulli model of the pipe has hinged ends. The inlet flow is considered constant steady that interacts with the wall of the pipe. The mathematical model is developed and its corresponding solution is obtained. The influence of the combination of variation of cross section, foundation stiffness and damping on the critical velocities, complex natural frequencies and stabilization of the system is presented.

Frequency response analysis of concrete seawall including soil-structure-seawater interaction

In this study, a frequency-domain finite element model is proposed to evaluate the seismic response of water-structure-sediment-foundation-backfill coupling system. It includes the seawater as a compressible fluid, seawall as viscoelastic solid, backfilled soil, sediment, and foundation as two-phase poroelastic domains with dynamic behavior described by u-p formulation of Biot’s equations. The dynamic interactions among various domains are fully considered and the PML is adopted as an absorbing boundary condition to truncate the computational domain, absorbing all out-going seismic waves. The dynamic responses in the structure, water and soil are assumed to be generated by the vertical propagating shear or compressional waves from the bedrock substrate under the foundation soil. Both the actions of horizontal and vertical seismic excitations at the bottom of foundation are respectively considered. After verifying the accuracy of PML and coupled model of soil-structure-water system, a series of parametric studies are conducted to investigate the effects of sediment thickness, water depth, foundation stiffness, foundation thickness, inclination of seawall, and properties of backfilled soil including soil permeability, shear modulus, and saturation degree, on the frequency response at the top of seawall.

On the wave propagation of the multi-scale hybrid nanocomposite doubly curved viscoelastic panel

In this paper, wave propagation analysis of multi-hybrid nanocomposite (MHC) reinforced doubly curved panel embedded in the viscoelastic foundation is carried out. Higher-order shear deformable theory (HSDT) is utilized to express the displacement kinematics. The rule of mixture and modified Halpin–Tsai model are engaged to provide the effective material constant of the MHC reinforced doubly curved panel. By employing Hamilton’s principle, the governing equations of the structure are derived and solved with the aid of an analytical method. Afterward, a parametric study is carried out to investigate the effects of the viscoelastic foundation, carbon nanotubes’ (CNTs’) weight fraction, various MHC patterns, radius to total thickness ratio, and carbon fibers angel on the phase velocity of the MHC reinforced doubly curved panel in the viscoelastic medium. The results show that, by considering the viscous parameter, the relation between wavenumber and phase velocity changes from exponential increase to logarithmic boost. A useful suggestion of this research is that the effects of fiber angel and damping parameter on the phase velocity of a doubly curved panel are hardly dependent on the wavenumber. The presented study outputs can be used in ultrasonic inspection techniques and structural health monitoring.

Static, dynamic and natural frequency analyses of functionally graded carbon nanotube annular sector plates resting on viscoelastic foundation

In this paper, static, dynamic and natural frequency responses of composite annular sector plate with carbon nanotubes (CNTs) reinforcements resting on viscoelastic foundation are investigated. The carbon nanotubes are considered to have uniform or functionally graded pattern in the plate thickness. Kelvin-voight model is used to model the viscoelastic foundation. To model the problem, Hamilton principle based on first shear deformation plate theory and finite element method are applied. The mechanical properties of annular sector plate composed of CNTs and polymer matrix are evaluated by using the rule of mixtures. The numerical results are obtained for investigating the effect of various factors such as distribution and volume fraction of CNTs, boundary conditions, stiffness and damping coefficients of viscoelastic foundation and sector angles on natural frequency, static and dynamic transient responses of the plate.