Nonlinear Elastic Foundation Research Articles

Fatigue Strength Analysis of Rail Fastening Baseplate

Superstructural elements of a railway track experience dynamic loads varying over time. As a consequence, this is accompanied by the initiation and accumulation of fatigue damages. Defects in intermediate rail fastening are quite difficult to identify using flaw detection methods, and some can be destructed. In particular, rail fastening baseplates often fail due to insufficient fatigue strength.Purpose: To investigate the fatigue strength of rail fastening baseplate.Methodology: Development of the mathematical model of deformation based on the finite element method (FEM). Strength analysis of the baseplate using the model of a variable-thickness slab resting on a nonlinear elastic foundation. The latter is modeled by a set of rods with nonlinear deformation and parameters determined by compression tests of the baseplate. A mathematical model developed in COSMOS software for the FE analysis of the stress-strain state of the baseplate at a time-varying load from rolling equipment and assembly forces from attaching the baseplate to the sleeper and rail.Research findings: The cyclic loading parameters are obtained at dangerous points of the baseplate and its durability is analyzed at different loads with respect to unevenness of track settlement and failure of bolt connections of rail fasteners. The influence of these factors on actual loads on the rail fastener is determined. It is found that the service life of the baseplate mainly depends on the dynamic load.

Nonlinear thermo-mechanical static stability analysis of FG-TPMS shallow spherical shells

An analytical solution for the nonlinear static stability problem of functionally graded triply periodic minimal surface (FG-TPMS) shallow spherical shells is studied in the current research for the first time. Three common TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) with three models of functionally graded porosity distribution along the thickness are considered. The shallow spherical shells (shallow SSs) are subjected to combined thermo-mechanical loadings and rested on a nonlinear elastic foundation. The fundamental formulas are expressed based on the higher-order shear deformation theory (HSDT) and von Kármán's geometrical nonlinearities. Employing the Ritz energy minimization method, the explicit relationship between load and deflection is derived. Subsequently, the static stability behavior of FG-TPMS shallow SSs is investigated. Numerical illustrations are investigated to show the superior thermo-mechanical load-carrying performance of the FG-TPMS SSs compared to corresponding isotropic structures of the same weight. The significant effects of geometric parameters, nonlinear elastic foundation parameters, and the type of FG-TPMS structures on the nonlinear static stability behavior of shallow SSs are further considered.

Application of bilateral iterative displacement control method to nonlinear free vibration analysis of dual-FG nanocomposite circular plates

This paper studies the geometrically nonlinear free vibration of a novel nanocomposite circular plate. The matrix of the nanocomposite structure is made of functionally graded (FG) polymer, and the distribution of graphene platelets (GPLs) as the reinforcement is assumed based on the five various linear FG models. Therefore, this novel nanocomposite is called dual-FG. The Voigt rule of mixture and Halpin-Tsai method are employed to homogenize FG polymer, and distribution of the GPLs within the matrix, respectively. Moreover, the whole structure is fixed on a nonlinear Kerr elastic foundation. The displacements of the plate are estimated by means of the first-order shear deformation theory (FSDT). Von-Kármán type of geometrically nonlinear relations expresses the plate's strains. The generalized differential quadrature (GDQ) technique, bilateral iterative displacement control method (BIDCM), and the weighted residual Galerkin procedure are implemented to extract the free vibration characteristics of the structure. The developed structure is validated with those reported in the open literature. A comprehensive parametric study is conducted to illustrate the effect of various parameters on the dynamic response of the dual-FG nanocomposite circular plate. It will be observed that the influence of the volume fraction and distribution model of GPL is more pronounced on the vibration frequency in the FG matrix. It was noted that the volume fraction of graphene platelets (GPL) and the power-law index of the matrix exhibit a direct and inverse relationship, respectively, with the natural frequency of the structure. Furthermore, the V-GPL model exhibits the highest percentage growth in nonlinear frequencies, while the Λ-GPL model shows the least growth. This observation is particularly pronounced for plates with hinged boundary conditions.

Large Amplitude Free Vibration of Elastically Restrained Tapered Beams Resting on Non-linear Elastic Foundation

Large Amplitude Free Vibration of Elastically Restrained Tapered Beams Resting on Non-linear Elastic Foundation

Nonlinear Buckling and Postbuckling Response of Porous FGM Shallow Spherical Caps and Circular Plates with Nonlinear Elastic Foundation Effects Using the Ritz Energy Method

Nonlinear Buckling and Postbuckling Response of Porous FGM Shallow Spherical Caps and Circular Plates with Nonlinear Elastic Foundation Effects Using the Ritz Energy Method

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Three-dimensional hygro-thermo-mechanical bending analysis of anisotropic skewed parallelogram plates on a tri-parameter foundation using a wavelet method

In the paper, a brand-new three-dimensional hygro-thermo-mechanical bending model of anisotropic skewed parallelogram plates with extreme skew angle in large deformation supported by a tri-parameter nonlinear elastic foundation is proposed. By investigating the mutual coupled interaction of hygrothermal effects between in-plane distribution and thickness variation, nine kinds of spatially analytical hygrothermal fields in parallelogram domain are established by analyzing the Dirichlet, Neumann and Robin thermal boundary conditions of sinusoidally induced heat conduction at top and bottom surfaces. With coordinate transformation in oblique system considering the Von Kármán membrane strains, the highly coupled and strongly nonlinear governing partial differential equations have been derived. A wavelet solving procedure has been developed with the convergence process validated and precision of obtained solutions verified in good accordance with published results. Parametric studies have been carried out to investigate the large-deflection behavior of simply supported or clamped skew plates on Winkler–Pasternak foundation under combined mechanical and hygrothermal loads. The influence of skew angle, material orthotropy and foundation contact nonlinearity is examined, which reveals the geometrically nonlinear responses of plates are highly affected by large oblique angle, principle orientation of material orthotropy and foundation parameters.

PINN-based forward and inverse bending analysis of nanobeams on a three-parameter nonlinear elastic foundation including hardening and softening effect using nonlocal elasticity theory

PINN-based forward and inverse bending analysis of nanobeams on a three-parameter nonlinear elastic foundation including hardening and softening effect using nonlocal elasticity theory

Application of pseudo-parameter iteration method to nonlinear deflection analysis of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation

In this study, the nonlinear deflection of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation was analyzed using the pseudo-parameter iteration method (PIM), which is a novel iterative semi-analytic method for solving ordinary/partial differential equations. To do this, we set six types of infinite beams with concave and convex shapes under static loading conditions. To calculate the nonlinear deflection of the infinite beam with variable cross-sections, the Bernoulli-Euler beam equation (fourth-order ordinary differential equation) considering changing beam flexural rigidity was introduced, and the PIM was adopted to this equation. Through the numerical experiment, it was confirmed that the nonlinear deflections calculated via the PIM are quite close to the exact solution within a few iterations. In addition, the graph of error quickly reaches the steady state error for all cases as the number of iterations increases.

Nonlinear damping and forced vibration investigation of three-layered viscoelastic sandwich beams on nonlinear elastic foundation with interlaminar continuous shear stress Zig–Zag theories

Nonlinear damping and forced vibration investigation of three-layered viscoelastic sandwich beams on nonlinear elastic foundation with interlaminar continuous shear stress Zig–Zag theories

Thermal effects on nonlinear vibration of nonlocal nanobeam embedded in nonlinear elastic medium

Nanotechnology has an impact on our lives in a many ways, from better medical treatments and more efficient energy sources to stronger and lighter materials and advanced electronics and this article presents the implementation of a perturbation method for the vibration analysis of simply supported and clamped–clamped Euler–Bernoulli nanobeams resting on nonlinear elastic foundations in thermal environment using nonlocal elasticity theory. Hamilton's principle is used to construct the differential equation of motion of a nanobeam in conjunction with appropriate boundary conditions. The equations of motion of the Euler–Bernoulli nanobeam are determined using nonlocal elasticity theory. It is shown how thermal loadings affect the vibration of the Euler–Bernoulli nanobeam. The multiple scale method, which is one of the perturbation method, is used to get an approximated solution for the presented system. The effects of temperature, Winkler, Pasternak and nonlinear foundation parameters on the vibration analysis of simply supported and clamped–clamped nanobeams are determined and results are given in tables and graphs.

On matched asymptotic expansion solutions of a bucklewave propagation problem

AbstractIn the present paper, the method of matched asymptotic expansions is used to derive approximate analytical solutions to a model problem related to bucklewave propagation in undersea pipelines. The model problem, introduced by Chater, Hutchinson and Neale (in: Collapse – The Buckling of Structures in Theory and Practice (Thompson, J.M.T., Hunt, G.W., eds.) Cambridge Univ. Press, Cambridge, 1983), consists of a linear elastic beam resting on a nonlinear elastic foundation, with inclusion of axial tension and inertia, related to a propagating buckle. When tension dominates over inertia, it is found that the asymptotic solution describing a propagating buckle has the character of nonoscillatory motion. When tension and inertia forces are of equal magnitude, or when inertia forces dominate over tension, the solutions are found to be oscillatory, but still localized around the bucklewave front. Finally, it is shown that when the propagation speed of the buckle exceeds a critical value, a spatial instability sets in, with exponentially growing amplitude of the beam deflection.

Sub-harmonic and super-harmonic resonances of spiral stiffened FG cylindrical panels of variable thickness embedded within a nonlinear elastic foundation

This research devotes to study the behaviors of sub- and super-harmonic resonances of spiral stiffened functionally graded (SSFG) variable thickness cylindrical panels (VTCP) exposed to external loading. The panel considered is rested on a nonlinear elastic foundation (NEF), which is composed of two linear foundations namely Pasternak and Winkler foundations, and a nonlinear foundation with cubic stiffness. The cylindrical panels of variable thickness are reinforced with internal spiral stiffeners. Both the cylindrical panels and spiral stiffeners are considered to be continuously graded along their thickness direction. To model the spiral stiffeners, the smeared stiffeners technique is utilized, and for deriving the discretized nonlinear governing equation of the SSFG-VTCP, the improved Donnell shell theory, von-Kármán equation, and Galerkin’s method are applied. In continuing, to examine the sub- and super-harmonic resonances, the method of multiple scales is implemented. Also, to validate the present results, in addition to comparing the present results with that of the previous research, the response of super-harmonic resonance, which is obtained theoretically, is compared with the 4th-order P-T method.

Nonlinear Oscillations of a Plate Resting on a Nonlinear Elastic Foundation and Forming the Bottom of a Plane Channel Filled with a Viscous Gas

This paper proposes a mathematical model for nonlinear oscillations of a Kirchhoff plate resting on an elastic foundation with hardening cubic nonlinearity and interacting with a pulsating layer of viscous gas. The plate is the bottom of a narrow plane channel filled with the viscous gas; the upper channel wall is rigid. Within this model, the aeroelastic response and phase response of the plate to pressure pulsation at the channel ends are determined and investigated. The formulated model allows us to simultaneously study the effect on the plate vibrations of its dimensions and the material physical properties, the nonlinearity of the plate elastic foundation, the inertia of gas motion, as well as the gas compressibility and its dissipative properties. The model was developed based on the formulation and solution of the nonlinear boundary value problem of mathematical physics. The equation of plate dynamics together with the equations

Numerical investigation on the thermal-nanofluidic flow induced transverse and longitudinal vibrations of single and multi-walled branched nanotubes resting on nonlinear elastic foundations in a magnetic environment

Carbon nanotubes (CNTs) have attracted great attention due to their tremendous mechanical, thermal, structural and electrical properties leading to many promising applications. The unique and interesting properties of CNTs, such as their mechanical and electrical features, have fascinated industries and researchers to implement CNTs for production of different electromechanical devices. However, it is established that the phenomenon of buckling often occurs when CNTs are subjected to compressive loads and their frequencies under vibration are in order of Tera-Hz and Giga-Hz. Moreover, the profound dynamic behaviours of nanotube under the influences of slip when conveying fluid and magnetic field require a serious and meticulous study. Therefore, this study focuses on the vibration analysis of nanofluid-conveying embedded multi-walled branched nanotubes resting on Winkler-Pasternak foundation in a thermal-magnetic environment. Using Euler-Bernoulli theory, Hamilton's principle and nonlocal elasticity theory, fully coupled equations of motion governing the transverse and longitudinal vibrations of the nanotube are developed. Also, the equation of the deformation of the nanotubes as well as the pressure variation in the tubes are developed. Additionally, Navier-Stoke's equation and energy equation for the fluid and nanotube are coupled with the vibration models. The dynamics of the multi-walled carbon nanotubes when coupled with Navier-Stokes and energy equations require meticulous study as it is different from the usual assumption of plug flow. Therefore, in this present work, the developed coupled systems of nonlinear partial differential equations are solved using multi-dimensional numerical PDEs solvers coupled with PDE-tools in MATLAB. With the aids of the solutions, parametric studies were performed. The results indicate that increasing the downstream angle decreases the stability of the system. Also, the results obtained from the dynamic behaviour of the system indicate that the magnetic effect has an attenuating or damping effect of about 20 %. Furthermore, the plug flow assumption deviates from actual working processes by over 11 %. The analytical solutions verified and validated with existing analytical, numerical and experimental results. It is envisaged that the present study will give better insight into nanotubes design and serve as an index for subsequent works in the research area.

Nonlinear Vibrations and Stability Analysis of GPL Reinforced Pipes Conveying Fluid Excited by Arbitrary Initial Conditions: An Optimized Analytical Solution

In this study, nonlinear vibrations and stability of multilayer functionally graded (FG) pipes conveying fluid laying on nonlinear elastic foundation and reinforced by graphene nanoplatelets (GPLs) are investigated with an emphasis on optimized analytical method. Various types of GPL distribution patterns are considered and to estimate the mechanical characteristics of the reinforced pipe the Halpin–Tsai micromechanics theory is applied. The nonlinear differential equation of motion is obtained by using Von–Kármán strain relations in conjunction with the Euler–Bernoulli beam theory and applying Hamilton’s principle. The Galerkin technique has been used for discretizing the partial differential equation. The optimized homotopy analysis method (HAM), as a strong analytical method, is utilized to solve the nonlinear differential equation by considering different initial excitations. The convergence-control parameter is taken into account to guarantee the convergence of the analytical solution. In this study, an exact solution based on HAM for the critical flow velocity and [Formula: see text]th nonlinear frequency is presented. Additionally, series solutions for the nonlinear time responses of the transverse and longitudinal vibrations of fluid-conveying pipe based on optimized HAM are obtained for the first time. In the numerical results, the effects of arbitrary initial conditions and different physical characteristics on the nonlinear frequencies and time responses are extensively examined. It is shown that the nonlinear frequencies and critical flow velocity of the pipe depend not only on the initial excitation amplitude, but also on the entire initial excitation function applied on the pipe.

Nonlinear phenomena in vibrations of embedded carbon nanotubes conveying viscous fluid

Various nonlinear phenomena such as bifurcations and chaos in the responses of carbon nanotubes (CNTs) are recognized as being major contributors to the inaccuracy and instability of nanoscale mechanical systems. Therefore, the main purpose of this paper is to predict the nonlinear dynamic behavior of a CNT conveying viscous fluid and supported on a nonlinear elastic foundation. The proposed model is based on nonlocal Euler–Bernoulli beam theory. The Galerkin method and perturbation analysis are used to discretize the partial differential equation of motion and obtain the frequency-response equation, respectively. A detailed parametric study is reported into how the nonlocal parameter, foundation coefficients, fluid viscosity, and amplitude and frequency of the external force influence the nonlinear dynamics of the system. Subharmonic, quasi-periodic, and chaotic behaviors and hardening nonlinearity are revealed by means of the vibration time histories, frequency-response curves, bifurcation diagrams, phase portraits, power spectra, and Poincaré maps. Also, the results show that it is possible to eliminate irregular motion in the whole range of external force amplitude by selecting appropriate parameters.

Nonlinear nonlocal damping effects under magnetic loads of a ferromagnetic-viscoelastic nanotube exposed to a nonlinear elastic medium with nonlocal viscosity

This study presents innovative results on a nonlinear dynamic model of a ferromagnetic-viscoelastic nanotube (FVN) whose dissipation effects are captured by material behavior with a linear dissipation law operating in a nonlinear geometric regime. It investigates a possible system derived from a Kelvin–Voigt type dissipation model that describes internal viscoelastic damping in the form of a parallel spring and a dashpot. In addition, a class of losses by linear viscous media in local and nonlocal damping is presented in the modeling. Magnetic load effects are studied using axial rod theory regardless of rotational inertia and shear effects. The governing equation is derived by Hamilton's principle for the FVN subjected to a transverse magnetic field resting on a nonlinear elastic foundation with viscose effects. Formulas have been developed to obtain the nonlinear damping characteristics of a single-walled nanotube. Then, multiple scale method (MSM)-based solutions and Eringen elasticity theory are applied to demonstrate the aforementioned effects on a narrow clamped-clamped (CC) nanotube in detail. The approximate analytical solution is verified by numerical integration methods and a good agreement between the approximate analytical and numerical results is observed. The effect of all important parameters such as local and nonlocal viscosity damping, nonlinear damping coefficient, linear and nonlinear stiffness of the elastic medium due to changes in amplitude vibration of the magnetic field (AVMF) and displacement responses (DR) have been investigated. The original research proposed in this paper may enable the systematic study of nonlinear damping in nanomechanical oscillators, which may help to reveal the underlying physical mechanism.

Nonlinear vibration of microbeams subjected to a uniform magnetic field and rested on nonlinear elastic foundation

Abstract This study investigates the nonlinear vibration motions of the Euler–Bernoulli microbeam on a nonlinear elastic foundation in a uniform magnetic field based on Modified Couple Stress Theory (MCST). The effect of size, foundation, and magnetic field on the nonlinear vibration motion of microbeam has been examined. The governing equations related to the nonlinear vibration motions of the microbeam are obtained by using Hamilton’s Principle, and the Multiple Time Scale Method was used to obtain the solutions for the governing equations. The linear natural frequencies of microbeam are presented in the table according to nonlinear parameters and boundary conditions. The linear and nonlinear natural frequency ratio graphs are shown. The present study results are also compared with previous work for validation. It is observed that length scale parameters and magnetic force have a more significant effect on the natural frequency of microbeams. It is seen that when the linear elastic foundation coefficient, the Pasternak foundation and the magnetic force effects increase, the ratio of nonlinear and linear natural frequency decreases.

Combined Effects of Surface Energy, Initial Stress and Nonlocality on Vibration of Carbon Nanotubes Conveying Fluid Resting on Elastic Foundations in a Thermo-Magnetic Environment

This paper scrutinizes the simultaneous impacts of surface elasticity, initial stress, residual surface stress and nonlocality on the nonlinear vibration of carbon nanotube conveying fluid resting while resting on linear and nonlinear elastic foundations and operating in a thermo-magnetic environment. The derived partial differential equation is decomposed into spatial and temporal equations using Galerkin method of decomposition. Thereafter, the temporal differential equation is solved with the aid of method of homotopy perturbation. Studies of the significance of the model parameters reveal that the negative value of the surface stress enhances the frequency ratio while the positive value of the surface stress abates the ratio. At any given value of nonlocal parameters, the surface effect is lessened for enhancing value of the length of the nanotube. The frequency ratio is abated as strength of the magnetic field, nonlocal parameter and the length of the nanotube are increased. The nonlocality lessens the surface effects and ratio of the frequencies. At high values of nonlocal parameter and nanotube length, the natural frequency of the structure gradually approaches nonlinear Euler–Bernoulli beam limit. The ratio of the frequencies is heightened when the temperature change is reduced at high temperature while at room/low temperature, such ratio is enhanced as the temperature change is augmented. Also, the frequency ratio at low temperatures is lower than at high temperatures. The present work will be very useful in the design and control of carbon nanotubes in thermo-magnetic environment while resting on elastic foundations.