Pasternak Elastic Medium Research Articles

The Model of Interconnected Pipes in the Investigation of Dynamic Stability of Carbon Nanotubes Embedded in an Elastic Medium

Abstract The dynamic stability of an interconnected single-walled carbon nanotube is investigated on the basis of the Euler-beam model and with the employment of the spectral Galerkin method. The tubes under investigation are assumed hinged at its both ends and are embedded in an elastic medium. In order to investigate the effect of the surrounding elastic medium on the dynamic stability of the pipe the Pasternak elastic medium is introduced. The critical velocity is calculated for different stiffness of the elastic medium and different density of the conveyed fluid.

Buckling Analysis of Orthotropic Graphene Sheets Supported on Pasternak Elastic Medium Using Nonlocal Elasticity Theory and Differential Quadrature Method

In this paper, the small scale effect on the buckling analysis of bi-axially compressed orthotropic Single-Layered Graphene Sheets (SLGS) supported on elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and supporting elastic medium. Using the principle of virtual work the governing differential equations are derived for rectangular orthotropic graphene sheets supported on elastic medium. Solutions for buckling loads for various boundary conditions are computed using Differential Quadrature Method (DQM). Parametric study has been performed to investigate the dependence of small scale effect on various graphene sheet parameters. It is observed that the nonlocal effect is significant in graphene sheets supported on elastic medium and has a decreasing effect on the buckling loads.

Nonlinear vibration and resonance analysis of a rectangular hyperelastic membrane resting on a Winkler–Pasternak elastic medium under hydrostatic pressure

In this paper, the nonlinear vibration and resonance analyses of a rectangular hyperelastic membrane embedded within a nonlinear Winkler–Pasternak elastic medium subjected to a uniformly distributed hydrostatic pressure are investigated. The material of the membrane is incompressible, homogeneous, isotropic, and hyperelastic. The constitutive law of neo-Hookean is utilized for modeling the system. The elastic foundation includes two Winkler and Pasternak linear terms and a Winkler term with cubic nonlinearity. Using the theory of thin hyperelastic membrane, Hamilton’s principle, and assuming the finite deformations, the governing equations are obtained. Then, the motion equation in the transverse direction is discretized by applying Galerkin’s method. The transverse nonlinear oscillations of the membrane are considered in two-modes. Then, utilizing the multiple scales method, the secondary resonance for the case of is analyzed. Also, to analyze the nonlinear vibration behavior, numerical methods including the fourth-order Runge–Kutta methods are utilized. In addition, in analyzing the nonlinear vibration responses of the rectangular hyperelastic membrane, the bifurcation diagram, quasi-period motion responses, and the phase portrait are examined. Finally, the influence of the geometrical characteristics and elastic foundation parameters on the resonance analysis of a rectangular hyperelastic membrane is investigated.

Mechanical Buckling of Circular Orthotropic Bilayer Nanoplate Embedded in an Elastic Matrix under Radial Compressive Loading

This article investigates the buckling behavior of orthotropic annular/circular bilayer graphene sheet embedded in Winkler–Pasternak elastic medium under mechanical loading. Using the nonlocal elasticity theory, the bilayer graphene sheet is modeled as a nonlocal orthotropic plate which contains small scale effect and van der Waals interaction forces. Differential Quadrature Method (DQM) is employed to solve the governing equations for various combinations of simply supported or clamped boundary conditions. The results show that small scale parameter does not have any effect on critical buckling load of cases without elastic medium in simply supported boundary condition. Also, increase of vdW coefficient leads to increase of critical buckling load smoothly then it has no impact on critical buckling load after a certain value.

Thermal Buckling Analysis of Circular Bilayer Graphene Sheets Resting on an Elastic Matrix Based on Nonlocal Continuum Mechanics

In this article, the thermal buckling behavior of orthotropic circular bilayer graphene sheets embedded in the Winkler–Pasternak elastic medium is scrutinized. Using the nonlocal elasticity theory, the bilayer graphene sheets are modeled as a nonlocal double–layered plate that contains small scale effects and van der Waals (vdW) interaction forces. The vdW interaction forces between the layers are simulated as a set of linear springs using the Lennard–Jones potential model. Using the principle of virtual work, the set of equilibrium equations are obtained based on the first-order shear deformation theory (FSDT) and nonlocal differential constitutive relation of Eringen. Differential quadrature method (DQM) is employed to solve the governing equations for simply-supported and clamped boundary conditions. Finally, the effects of the small scale parameter, vdW forces, aspect ratio, elastic foundation, and boundary conditions are considered in detail.

Effect of magnetic-thermal field on nonlinear wave propagation of circular nanoplates

ABSTRACTIn this article, an analytical approach is developed to investigate the nonlinear vibrational behavior of an graphene sheet. In order to be a practical instance, the circular nanoplate is also resting on an external Pasternak elastic medium including the Winkler modulus and shear modulus parameters. At the same time, it is subjected to the magnetic field and a thermal load. The modified couple stress theory as well as Eringen nonlocal elasticity theory involving small scale parameters are implemented to justify the size-dependent effects. The governing equations are derived based on von Kármán nonlinear strain–displacement relations. The presented method is verified by comparing the obtained results to their counterparts reported in the published literature. Effects of various parameters such as nonlocal parameters, temperature change, and elastic medium coefficients for different boundary conditions are also discussed. Numerical results reveal that the intrinsic size-dependent property of material increases the natural frequency.

Analysis of buckling of a multi-layered nanocomposite rectangular plate reinforced by single-walled carbon nanotubes on elastic medium considering nonlocal theory of Eringen and variational approach

In this paper, the mechanical buckling of a multi-layered nanocomposite rectangular plate reinforced by single-walled carbon nanotubes (SWCNTs) on Pasternak elastic medium subjected to in-plane loadings is investigated. The equilibrium and stability equations are derived using the higher-order shear deformation plate theory (HSDT) considering the nonlocal theory of Eringen and variational approach. Also, uniformly distributed carbon nanotubes (CNTs) plate is considered. Differential quadrature method (DQM) is used for solving the governing equations for various boundary conditions. In the current study, the effects of nonlocal parameters, aspect ratios, boundary conditions and medium modulus on the buckling behavior of nanocomposite plate are studied. Finally, the results of simpler examples are verified by other references. It is observed that the increase in nonlocal parameter leads to a decrease in non-dimensional buckling load, and as the scale factor increases, the nonlocal parameters enhance, and by increasing the nonlocal parameter, buckling load reduces.

A Unified Higher-Order Beam Theory for Free Vibration and Buckling of FGCNT-Reinforced Microbeams Embedded in Elastic Medium Based on Unifying Stress–Strain Gradient Framework

The main object of this research is to formulate the linear free vibration as well as static stability of embedded functionally graded carbon nanotube-reinforced composite microbeams in thermal environment. The nonlocal stress–strain gradient theory in conjunction with the unified higher-order beam theory by considering the temperature dependence of material properties and the initial thermal stresses is used to derive nonclassical governing equations. The eigenvalue problems governing the linear vibration and static stability of microbeams are obtained by using the weak form of partial differential equations and employing Chebyshev–Ritz method. The fast rate of convergence of the method is demonstrated numerically, and its accuracy is verified by comparing the results in the limit cases with existing solutions in the literature. The effects of transverse shear stress distribution along the thickness together with the spring constants of Winkler–Pasternak elastic medium, different distribution patterns of CNTs across the thickness, the temperature dependence of material properties, the temperature rise, boundary conditions, nonlocal stress and strain gradient parameters on the frequency parameters and load-bearing capacity are investigated. Findings show that the effects of Pasternak constant of elastic medium on the natural frequency as well as critical buckling load depend on the boundary conditions. It is also shown that the nonlocal stress and strain gradient parameters have opposite effects on the stiffness. The more effective distribution pattern of CNTs across the thickness which enhances the static stability and vibratory behavior of microbeam is determined as well.

Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects

Based on nonlocal Timoshenko beam theory, dynamic stability of functionally graded (FG) nanobeam under axial and thermal loading was investigated. Surface stress effects were implemented according to Gurtin-Murdoch continuum theory. Using power law distribution for FGM and von Karman geometric nonlinearity, governing equations were derived based on Hamilton's principle. The developed nonlocal models have the capability of interpreting small scale effects. Pasternak elastic medium was employed to represent the interaction of the FG nanobeam and the surrounding elastic medium. A parametric study was conducted to focus influences of the static load factor, temperature change, gradient index, nonlocal parameter, slenderness ratio, surface effect and springs constants of the elastic medium on the dynamic instability region (DIR) of the FG beam with simply-supported boundary conditions. It was found that differences between DIRs predicted by local and nonlocal beam theories are significant for beams with lower aspect ratio. Moreover, it was observed that in contrast to high temperature environments, at low temperatures, increasing the temperature change moves the origin of the DIR to higher excitation frequency zone and leads to further stability. Considering surface stress effects shifts the DIR of FG beam to higher frequency zone, also increasing the gradient index enhances the frequency of DIR.

Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory

The modified couple stress-based third-order shear deformation theory is presented for sigmoid functionally graded materials (S-FGM) plates. The advantage of the modified couple stress theory is the involvement of only one material length scale parameter which causes to create symmetric couple stress tensor and to use it more easily. Analytical solution for dynamic instability analysis of S-FGM plates on elastic medium is investigated. The present models contain two-constituent material variation through the plate thickness. The equations of motion are derived from Hamilton\'s energy principle. The governing equations are then written in the form of Mathieu-Hill equations and then Bolotin\'s method is employed to determine the instability regions. The boundaries of the instability regions are represented in the dynamic load and excitation frequency plane. It is assumed that the elastic medium is modeled as Pasternak elastic medium. The effects of static and dynamic load, power law index, material length scale parameter, side-to-thickness ratio, and elastic medium parameter have been discussed. The width of the instability region for an S-FGM plate decreases with the decrease of material length scale parameter. The study is relevant to the dynamic simulation of micro structures embedded in elastic medium subjected to intense compression and tension.

Nonlocal buckling analysis of functionally graded annular nanoplates in an elastic medium with various boundary conditions

In this article, buckling of functionally graded (FG) single-layered annular graphene sheets embedded in a Pasternak elastic medium is investigated using the nonlocal elasticity theory. The material properties of the FG graphene sheets are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. Using the principle of virtual work, the governing equations are derived based on first-order shear deformation theory and the nonlocal differential constitutive relations of Eringen. Differential quadrature method is also utilized to solve the equilibrium equations for various combinations of free, simply supported and clamped boundary conditions. In order to assure the accuracy of the results, convergence properties of the critical buckling load are examined in detail. To verify the present study, some comparison studies are carried out between the obtained results and the available solutions in the literature. A parametric study is then conducted to investigate the influences of small scale effects, grading index, surrounding elastic medium, boundary conditions, buckling mode and geometrical parameters on the critical buckling load.

Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory

In this paper, size-dependent free vibration analysis of curved functionally graded nanobeams embedded in Winkler–Pasternak elastic medium is carried out via an analytical solution method. Three kinds of boundary condition namely, simply supported-simply supported, simply supported-clamped and clamped-clamped are investigated. Material properties of curved functionally graded beam change in thickness direction according to the Mori–Tanaka model. Nonlocal strain gradient elasticity theory is adopted to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Nonlocal governing equations of curved functionally graded nanobeam are obtained from Hamilton’s principle based on Euler–Bernoulli beam model. Finally, the influences of length scale parameter, nonlocal parameter, opening angle, elastic medium, material composition, slenderness ratio and boundary conditions on the vibrational characteristics of nanosize curved functionally graded beams are explored.

Nonlinear thermo-elastic bending behavior of graphene sheets embedded in an elastic medium based on nonlocal elasticity theory

This paper investigates the large deflection behavior of orthotropic single layered graphene sheet (SLGS) embedded in a Winkler–Pasternak elastic medium under a uniform transverse load in thermal environments. Using the nonlocal differential constitutive relations of Eringen, the SLGS is modeled as a nonlocal orthotropic plate. Using the principle of virtual work, the coupled nonlinear equilibrium equations are obtained based on first-order shear deformation theory (FSDT) and the von Kármán geometrical model. The differential quadrature (DQ) discretized form of the governing equations with clamped and simply supported boundary conditions is derived. The Newton–Raphson iterative scheme is used to solve the resulting system of nonlinear algebraic equations. Effects of small scale parameter, thermal environment, width-to-length elasticity ratio, aspect ratio, thickness, elastic foundation, load value and boundary conditions are considered in detail. The results show that, unlike the simply supported boundary conditions, increase of small scale parameter plays a decreasing role in effect of thermal environment on the deflections of nanoplates with clamped edges. It is also observed that increasing the plate thickness, thermal load effects decline noticeably and depending on the small scale value the thermal loads do not have any significant effect on the results beyond a specified thickness.

Thermo-electro-mechanical vibration analysis of size-dependent nanobeam resting on elastic medium under axial preload in presence of surface effect

In the present manuscript, a nonclassical beam theory is developed to analyze free vibration of piezoelectric nanobeam by considering surface effects resting on Winkler–Pasternak elastic medium and thermal loading with axial preload. The nonclassical Eringen theory is utilized to incorporate the length-scale parameter to account for the small-scale effect, while the Gurtin–Murdoch model is employed to inject the surface effects including surface elasticity, surface stress and surface density. The governing equations are derived using Hamilton’s principle in the framework of Euler–Bernoulli beam theory. The governing partial differential equations of motions of system are reduced to a set of algebraic equations with the help of differential transformation method as a semi-analytical–numerical. The mathematical derivations and numerical results are presented in detail for various boundary conditions. Some numerical examples are illustrated in order to investigate the effect of several parameters such as the nonlocal parameter, piezoelectric voltage, surface effects, temperature change, axial preload and elastic medium parameters. Moreover, it is also indicated that the numerical results have good agreement with previous studies.

Using the modified strain gradient theory to investigate the size-dependent biaxial buckling analysis of an orthotropic multi-microplate system

On the basis of the modified strain gradient theory, this research presents an analytical approach to analyze elastic instability of an orthotropic multi-microplate system (OMMPS) embedded in a Pasternak elastic medium under biaxial compressive loads. Kirchhoff plate theory and the principle of total potential energy are applied to obtain the partial differential equations and corresponding boundary conditions. Various types of “chain” boundary conditions for the ends of the microplates system are assumed such as “Clamped-Chain,” “Free-Chain” and “Cantilever-Chain” systems. In order to analytically obtain the buckling load of the OMMPS, we use Navier’s approach which satisfies the simply supported boundary conditions and trigonometric method. In order to show the dependability of the presented formulation in this paper, several comparison studies are carried out to compare with existing data in the literature. Numerical results are presented to reveal variations of the buckling load of OMMPS corresponding to various values of the number of microplates, the length scale parameter \({\left({\frac{h}{l}}\right)}\), aspect ratio, Pasternak elastic medium parameters and the thickness of the microplate and the biaxial compression ratio. Some numerical results of this paper illustrate that when the number of microplates is small, especially becoming 2, there is an important difference between buckling loads obtained for “Clamped-Chain,” “Free-Chain” and “Cantilever-Chain” systems. Also, it is shown that by increasing the number of microplates in the system, the influence of the Pasternak elastic medium on the buckling load of system is reduced. It is anticipated that the results reported in this work are applied as a benchmark in future microstructure issues.

Study the Surface Effect on the Buckling of Nanowires Embedded in Winkler–Pasternak Elastic Medium Based on a Nonlocal Theory

Nano structures such as nanowires, nanobeams and nanoplates have been investigated widely for their innovative properties. In this paper the buckling of nanowires surrounded in a Winkler - Pasternak elastic medium has been examined based on the nonlocal Euler-Bernoully model with considering the surface effects. In the following a parametric study that explores the influence of numerous physical and geometrical parameters on the buckling of nanowires is presented. It has been shown that by growing the ratio of surface area to bulk in nano-size structures, the effect of surface energy turn out to be important and should be taken into consideration. Moreover the results point out that surface elasticity and residual surface tension stimulus the buckling behavior of nanowires.

A four-variable refined plate theory for dynamic stability analysis of S-FGM plates based on physical neutral surface

Dynamic instability analysis of S-FGM on elastic medium is investigated using a four-variable refined plate theory. The material is graded in the thickness direction and two power law based on the rule of mixture is used to estimate the effective material properties. The neutral surface position for such S-FGM plates is determined and the third-order shear deformation theory based on exact neutral surface position is employed. If the neutral surface is used, the FGM plates can easily be treated, because the coupling between stretching and bending deformations is not occurred. Even though present theory uses only four unknown variables, have strong similarities with CPT in many aspects such as boundary conditions, equation of motion and stress resultant expressions. Hamilton’s principle is utilized to derive the equations of motion. It is assumed that the elastic medium is modeled as Pasternak elastic medium. The governing equations are written in the form of Mathieu–Hill equations and then Bolotin’s method is employed to determine the instability regions. Various numerical results are presented to reveal the influences of static and dynamic load factors, power law index, elastic medium parameter, and side-to-thickness ratio on the dynamic stability characteristics of S-FGM plates. Also, the formulation should provide engineers with the capability for the design of S-FGM plates for special technical applications including aircraft, rocket, wing surfaces and missile skins.

Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory

In this article, non-uniform biaxial buckling analysis of orthotropic single-layered graphene sheet embedded in a Pasternak elastic medium is investigated using the nonlocal Mindlin plate theory. All edges of the graphene sheet are subjected to linearly varying normal stresses. The nanoplate equilibrium equations are derived in terms of generalized displacements based on first-order shear deformation theory (FSDT) of orthotropic nanoplates using the nonlocal differential constitutive relations of Eringen. Differential quadrature method (DQM) has been used to solve the governing equations for various boundary conditions. The accuracy of the present results is validated by comparing the solutions with those reported by the available literatures. Finally, influences of small scale effect, aspect ratio, polymer matrix properties, type of planar loading, mode numbers and boundary conditions are discussed in details.

Bending and vibration analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory

Abstract A model for sigmoid functionally graded material (S-FGM) microplates based on the modified couple stress theory with first order shear deformation is developed. The advantages of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material׳s micro-structural characteristics. The present model of microplate can be viewed as a simplified couple stress theory in engineering mechanics. The present models contain one material length scale parameter and can capture the size effect, and two-constituent material variation through the plate thickness. The equations of motion are derived from Hamilton׳s principle based on the modified couple stress theory, and the power law variation of the material through the thickness of the plate. Material properties of functionally graded plate are assumed to vary according to two power law distribution of the volume fraction of the constituents. The elastic medium is modeled as Pasternak elastic medium. Analytical solution of rectangular plates is derived, and the obtained results are compared well with reference solutions. Finally, the influences of power law index, material length scale parameter, thickness ratio, and elastic medium parameter on the deflection and the natural frequency of plates have been investigated.

Thermo-electro-mechanical vibration of coupled piezoelectric-nanoplate systems under non-uniform voltage distribution embedded in Pasternak elastic medium

In the present paper, the thermo-electro-mechanical vibration characteristics of a piezoelectric-nanoplate system (PNPS) embedded in a polymer matrix are investigated. The system is subjected to a non-uniform voltage distribution. The voltage distribution and in-plane preloads are very important in the resonance mode of smart composite nanostructures using PNPS. Small scale effects are taken into consideration using the nonlocal continuum mechanics. Hamilton's principle is employed to derive the nonlocal equations of motion. The governing equations are solved for various boundary conditions by using differential quadrature method (DQM). To verify the accuracy of the present results, a closed-form solution is also derived for the natural frequencies of simply supported PNPSs. The results of DQM are compared with those of exact solution and an excellent agreement is found. Finally, the effects of initial preload, temperature change, boundary conditions, aspect ratio, length-to-thickness ratio, nonlocal and non-uniform parameters on the vibration characteristics of PNPSs are studied. It is shown that the natural frequencies are quite sensitive to the non-uniform and nonlocal parameters.