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Folded structures hold significant importance in aerospace technology due to their distinct design, effectively reconciling strength with a lightweight framework. Within aerospace applications, these structures fulfill diverse roles, serving as pivotal elements in aircraft wings, fuselage sections, and essential components of spacecraft and satellites. This study delves into the free vibration behavior of folded sandwich plates composed of a porous core and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings, resting on a two-parameter elastic foundation. Leveraging first-order shear deformation plate theory (FSDT), elasto-dynamic equations are established for each individual plate segment. Utilizing the two-dimensional generalized differential quadrature method (2D-GDQM), these equations are discretized. Subsequently, by applying boundary conditions to all six edges, including traditional clamped, simply supported, and free configurations, and incorporating continuity conditions for displacements, forces, and moments at the joint borders, natural frequencies and modal shapes are extracted for the folded plate structure. The validity of the proposed formulation and results is demonstrated by comparing them with numerical examples for both composite flat and folded plates. This comparison provides strong support for the accuracy and reliability of the present study. The study comprehensively examines the influence of various parameters on natural frequencies, including modal shape visualizations under different boundary configurations, plate thickness, crank angle, porosity parameter, core thickness, foundation stiffness, and CNTs distribution type. [ABSTRACT FROM AUTHOR]

This work deals with the wave propagation analysis in functionally graded carbon nanotubes (CNTs)-reinforced composite beams lying on an elastic medium. Despite the large amount of experimental and theoretical studies in the literature on the mechanical behavior of composite structures strengthened with CNTs, limited attention has been paid to the effect of an axial graduation of the reinforcing phase on the mechanical response of CNTs-reinforced composite beams. In this paper, CNT fibers are graded across the beam length, according to a power-law function, which expresses a general variation from a linear to parabolic pattern. An Euler-Bernoulli beam theory is considered herein to model the CNTs-reinforced composite structure resting on a Winkler–Pasternak foundation, whose governing equations are derived from the Hamiltonian principle. The theoretical solution of the problem checks for the sensitivity of the mechanical response to different parameters, i.e., the wave number, power index, Winkler and Pasternak coefficients, that could serve for further computational/experimental studies on the same problem, even from a design standpoint. [ABSTRACT FROM AUTHOR]

We consider an unsteady elastic diffusion vibration problem of an orthotropic Timoshenko beam on an elastic foundation under a distributed transverse load. The Winkler model is used as an elastic foundation model. We use the system of Timoshenko beam bending equations taking into account diffusion for the mathematical problem formulation. These equations are obtained with the d'Alembert variational principle applied to the elastic diffusion continuum model. The resulting model considers the diffusion fluxes relaxation. The problem solution is sought as convolutions of Green's functions with functions defining unsteady distributed disturbances. The integral Laplace transform in time and the expansion in the Fourier series in the longitudinal coordinate are used to find the Green's functions. A calculation example for the beam with a rectangular section is considered. The beam deflections and the concentration increments under the action of an impulsively applied distributed transverse load are found. Finally, the main conclusions about the coupling effect of the stress–strain state and mass transfer in the beam are represented. [ABSTRACT FROM AUTHOR]

Efficient and accurate simulation of the vibration characteristics of a composite pipeline system is the key to the study of the stability and vibration control of the pipeline system. A simulation method called transfer matrix method for multibody systems is used to predict the vibration of a composite pipeline resting on an elastic soil. The transfer matrix of the Euler–Bernoulli beams considering the internal fluid velocity and high-efficiency dynamics model of the pipeline system under the action of the elastic foundation are derived. The simulation results have good agreement with that of the literature and commercial software ANSYS Workbench which verified the accuracy of the numerical model. The simulation results show that with the increase of the velocity, the natural frequencies of each mode of the pipeline decrease continuously. When the first frequency is zero, the pipeline buckling occurs and the velocity reaches the critical velocity; the elastic coefficient and shear coefficient in the foundation coefficient are positively related to the stability of the pipeline system. The damping coefficient is negatively related to pipeline stability. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the buckling behavior of porous Functionally Graded Materiel (FGM) cylindrical shells based on Donnell shell theory. In this context, we develop an explicit analytical expression which takes into consideration the effect of porosities through the thickness of the structure and that of the elastic foundation using a modified power-law function and the models of Winkler and Pasternak, respectively. We use the modified rule of mixture to determinate the behavior of the porous FGM cylindrical shell. The effects of porosity volume fraction, power-law index, and Young's modulus ratio are investigated. Moreover, we also discuss the influence of different parameters on the stability behavior of the porous FGM shell. [ABSTRACT FROM AUTHOR]

In this paper, the free vibration of the functionally graded porous (FGP) non-uniform annular-nanoplates lying on Winkler foundation (WF) is studied by using the smoothed finite element method based on the first-order shear deformation theory (FSDT). The combination of the mixed interpolation of the tensorial components for the three-node triangular element (MITC3 element) and the edge-based smoothed finite element method (ES-FEM) creates the ES-MITC3 element. This element is employed to avoid the shear locking problem as well as to improve the accuracy of the original MITC3 element. The small-scale effect is considered based on the nonlocal theory. Applying Hamilton's principle, the governing equation of the FGP non-uniform thickness annular-nanoplate is derived. Material properties of the nanoplate are characterized by two parameters: power-law index (k) and maximum porosity distributions (Ω) in the forms of cosine functions. The results of the present work are compared with other published work to verify accuracy and reliability. Moreover, the effects of geometry parameters and material properties on the free vibration of FGP non-uniform annular-nanoplates are comprehensively investigated. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the effect of variable foundation stiffness on the dynamic response of an infinite railway track under the action of a harmonic moving load. In this study, harmonic variation in the foundation stiffness along the track length is considered. Here, the dynamic response of the finite element (FE) model is obtained with the help of Newmark Beta method using programming in MATLAB. It is ascertained that in the central region of the long FE model, the response is repetitive, thereby ensuring that boundary conditions do not influence the response in the central region. In this problem, two frequencies, i.e. frequency of moving load and the spatial frequency of variable foundation stiffness, are involved. Their combined influence on the dynamic characteristics such as resonant frequency, critical velocity, displacement time history, displacement below load, and bifurcation curve are investigated. It is shown that the dynamic response is qualitatively and quantitatively affected by the wavelength (λ) and amplitude (ε) of the variation of foundation stiffness. It is also shown that the important dynamic properties, i.e., the critical velocity and the resonant frequency reduce with the increase in wavelength of stiffness variation. This reduction is significant for the large amplitude of harmonic stiffness variation. [ABSTRACT FROM AUTHOR]

Different numerical schemes are introduced to formulate and solve nonlinear vibration analysis of elastically supported multilayer composite plate problems. The type of elastic foundation is Winkler - Pasternak foundation model. The governing equations are formulated according to a first order transverse shear theory. Examined schemes are based on polynomial, sinc, discrete singular convolution differential quadrature (DSCDQ) methods. These schemes are appointed to reduce the problem to nonlinear Eigen-value problem. The reduced system is solved iteratively. Numerical analysis is performed to explore influence of different computational characteristics on convergence and efficiency of the obtained results. Comprehensive numerical results validate the solutions by comparison with those obtained by the exact and numerical ones. Moreover, a parametric study is introduced to investigate the influence of supporting conditions, foundation parameters, elastic and geometric characteristics of the vibrated plate, on natural frequencies and mode shapes. The obtained results exhibit that the (DSCDQM) is an accurate efficient method in the dynamic analysis of discontinuity plates resting on nonlinear elastic foundation. [ABSTRACT FROM AUTHOR]

In order to meet the needs of vibration reduction in complex vibration environments, one-dimensional phononic crystals constituted with periodic tunnels on elastic foundations are proposed. The governing equations of the flexural wave in a periodic structure are established based on Euler beam theory and its band structure is calculated by the plane wave expansion method. The theoretical and numerical results show that there are distinct forbidden bands and pass bands when the flexural wave propagates in the periodic tunnel structure. The elastic foundation stiffness, elastic modulus ratio, and axial prestress have significant effects on the location and width of the band gap. The results of the paper could be used for the vibration mitigation of the tunnel structure. [ABSTRACT FROM AUTHOR]

An analytical approach is developed in this paper to investigate the wave propagating in graphene oxide powder (GOP) reinforced nanocomposite plates under thermal loading. The plate is embedded on a two parameter elastic substrate to make this research more realistic. The structure is exposed to thermal loadings with various temperature rises such as sinusoidal temperature rise (STR), linear temperature rise (LTR) and uniform temperature rise (UTR). The governing differential equations are derived via incorporation of the Hamilton's principle and refined higher order plate theory then solved analytically to obtain the wave frequency and phase velocity of the GOP reinforced nanocomposite plate. The GOPs are dispersed in a polymeric matrix through the thickness of the plate by considering different functionally graded (FG) patterns. Moreover, the effective material properties of the structure are predicted on the basis of Halpin–Tsai micromechanical scheme. Furthermore, the accuracy of the present model is verified with previous works and also the influences of various parameters on the wave propagation behavior of the structure are covered too. The graphical results declare that wave propagation in GOP reinforced nanocomposite plates dramatically depends on these parameters. [ABSTRACT FROM AUTHOR]

In this paper, a functionally graded piezoelectric plate on an elastic foundation composed of two different piezoelectric materials bonded together in the form of plate is studied, and its static bending and fundamental frequencies are analyzed. First, based on Kirchhoff plate theory and the Hamilton principle, the governing equations and corresponding boundary conditions of the model are derived, and then the equations are discretized and solved by the differential quadrature method (DQM). Finally, the effects of physical parameters such as length-to-height ratio, length-to-width ratio, material graded index, foundation stiffness coefficient, temperature change value and external voltage value on static bending deflection, and fundamental frequency value of the functionally graded piezoelectric plate with four sides simply supported are discussed. The calculated results are in good agreement with those in the literature. The data results show that the increase in the elastic foundation stiffness coefficient will increase the equivalent stiffness of the plate. In the process of work, due to the equivalent pressure value generated by the influence of the external voltage, it will lead to unstable phenomena. [ABSTRACT FROM AUTHOR]

This article proposes a refined higher order nonlocal strain gradient theory for stresses and deflections of new model of functionally graded (FG) sandwich nanoplates resting on Pasternak elastic foundation. Material properties of the FG layers are supposed to vary continuously through-the-thickness according to a power function or a sigmoid function in terms of the volume fractions of the constituents. The face layers are made of FG material while the core layer is homogeneous and made of ceramic. In this study, an analytical approach is proposed using the higher-order shear deformation plate theory and nonlocal strain gradient theory with combination of various boundary conditions. Numerical outcomes are reported to display the impact of the material distribution, boundary conditions, elastic foundation parameters and the sandwich nanoplate geometry on the deflections and stresses of FG sandwich nanoplates. The exactness of this theory is determined by comparing it to other published outcomes.

In this paper, we analyze the effects of the time-delay feedback position on the dynamical behavior of the nonlinear beam on elastic foundation under periodic external force and the chaos control. Firstly, by using the formulation of Lagrange as well as geometric analysis, the nonlinear equation of the system with time-delay is established. In this case, the system rests on a Winkler-type elastic foundation soil that acts on its bottom interface (soil-beam). Secondly, the equilibrium points are found, their stability studied and the condition of Hopf bifurcation established. Thirdly, by using the Melnikov process, the analytical constraints necessary to have chaos or not as a behavior of the system are sought, which is confirmed by numerical investigations, with plotting of the time series, Lyapunov exponent and the bifurcation diagram, which are used to study the effects of time-delay on dynamical behavior of the system. This time-delay being used as a control parameter. The results show that when the value of time delay is small, the ergodic tori and resonance cycles with different rotation numbers on the torus can be observed. When the value of time-delay increases the chaos behaviors take place in the system. Taking into account the fact that chaotic situations are considered disturbance and are harmful for operation, the control method based on the time-delay parameter is used to quench and also to prevent chaotic behaviors.

This paper uses the finite element method to simulate the mechanical, electric, and polarization behaviors of piezoelectric nanoplates resting on elastic foundations subjected to static loads, in which the flexoelectric effect is taken into consideration. The finite element formulations are established by employing a new type of shear deformation theory, which does not need any shear correction factors, but still accurately describes the stress field of the plate. The numerical results show clearly that the flexoelectric effect has a strong influence on the mechanical responses of the nanoplates. In particular, the normal stress distribution in the thickness direction is no longer linear when the flexoelectric coefficient is large enough, and this phenomenon differs completely from that of conventional plates. In addition, the distribution of the electric field and the polarization also depend on boundary conditions, which were not investigated in the published works. [ABSTRACT FROM AUTHOR]

Nonlinear buckling analysis for honeycomb auxetic-core sandwich toroidal shell segments with CNT-reinforced face sheets surrounded by elastic foundations under the radial pressure is presented in this study. The basic equation system of shells is established based on the von Kármán–Donnell nonlinear shell theory, combined with Stein and McElman approximation. Meanwhile, the foundation-shell elastic interaction is simulated by the foundation model based on the Pasternak assumption. The Galerkin procedure is utilized to achieve the pre-buckling and post-buckling responses for the shell, from which the radially critical buckling load is determined. Numerical analysis shows the various influences of auxetic-core layer, CNT-reinforced face sheets, and elastic foundation on the pre-buckling and postbuckling behavior of sandwich shells with CNT reinforced face sheets. [ABSTRACT FROM AUTHOR]

The main interest of this paperwork is to examinate the dynamic behavior (free vibrational response) of carbon nanotubes (CNT) composite beams standing on an elastic foundation of Winkler-Pasternak's. The affected beam consists of a polymer matrix reinforced with single-wall carbon nanotubes (SWCNT's), in which, a large number of CNT's reinforcement of infinite length are distributed in a linear elastic polymer matrix. In this study the CNT's are considered either aligned or randomly oriented on the matrix. A refined high-order beam theory (RBT) is adopted in the present analysis using a new shape function. The refined beam theory which is summarized by differentiating the displacement along the beam transverse section into shear and bending components, initially the material properties of the composite beam (CNTRC) are estimated using the Mori-Tanaka's method. The beam is considered simply supported on the edge-lines. NAVIER's solutions are proposed to solve the boundary conditions problems. Since there are no results to compare with in the literature; the results in this study are compared with a free vibrational analysis of an isotropic beam. Several aspects such as the length/thickness ratio, volume fraction of nanotubes, and vibrational modes are carried out in the parametric study. [ABSTRACT FROM AUTHOR]

This present study consists to analyze the mechanical buckling and the free vibration stabilities of antisymmetric cross-ply and angle-ply laminated composite plates using a refined high order shear deformation theory of four variables against five in other high order theories. Among the advantages of this new theory it takes into consideration the shearing effect in the calculation of deformation without the need for shear correction factors and giving rise to a variation of the shear stresses along the thickness and satisfying the zero shear stresses condition in faces of the plate. The laminate resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The equations of the motion are derived from Hamilton's principal. The closed form solution of simply supported rectangular plates has been obtained by using the Navier method. In addition, the effects of various parameters of the laminated composite plate on static buckling and dynamic are presented. [ABSTRACT FROM AUTHOR]

In this study, free vibration and buckling behaviors of a functionally graded nanoplate supported by the Winkler–Pasternak foundation using a nonlocal classical plate theory are investigated. Eringen's nonlocal differential model has been used for considering the small-scale effect. The properties of the functionally graded nanoplate are considered to vary transversely following the power law. The governing vibration and buckling equations of an elastically supported functionally graded nanoplate have been derived using the principle of virtual work, and the solution is obtained using the Rayleigh–Ritz method and characteristic polynomials. The advantage of this method is that it disposes of all the drawbacks regarding edge constraints. The objective of the article is to see the effect of edge constraints, aspect ratios, material property exponent, nonlocal parameter, and foundation parameters on the nondimensionalized frequency and the buckling load of an embedded functionally graded nanoplate in a thermal environment. The study highlights that the nonlocal effect is pronounced for higher modes and/or higher aspect ratios and need to be considered for the analysis of the nanoplate. Further, it is observed that the effect of the Pasternak foundation is prominent on nondimensionalized frequencies and buckling of the functionally graded nanoplate. [ABSTRACT FROM AUTHOR]

Rock masses with a distinct structure may present a transversely isotropic character; thus, the stress state in a transversely isotropic elastic half-plane surface is an important way to assess the behavior of the interaction between the distributed loading and the surroundings. Most previous theoretical analyses have considered a loading direction that is either vertical or horizontal, and the stress distribution that results from the effect of different loading directions remains unclear. In this paper, based on the transversely isotropic elastic half-plane surface theory, a stress solution that is applicable to distributed loading in any direction is proposed to further examine the loading effect. The consistency between the analytical solution and numerical simulations showed the effectiveness of the proposal that was introduced. Then, it was utilized to analyze the stress distribution rule by changing the Poisson's ratio and Young's modulus of the model. The effects of the formation dip angle on the stress state are also examined. The stress distribution, depending on the physical property parameters and relative angle, is predicted using an analytical solution, and the mechanisms associated with the transversely isotropic elastic half-plane surface subjected to the loading in any direction are clarified. Additionally, extensive analyses regarding this case study, with respect to the mechanical behavior associated with changes in the stress boundary, is available. Hence, the proposed analytical solution can more realistically account for the loading problem in many engineering practices. [ABSTRACT FROM AUTHOR]

Abstract For arbitrary two-point boundary condition, which makes the corresponding linear uniform problem well-posed, we obtain an existence and uniqueness result for the boundary value problem of finite beam deflection resting on arbitrary nonlinear non-uniform elastic foundation. The difference between the desired solution and the corresponding linear uniform one in L ∞ $L^{\infty}$ sense is bounded explicitly in terms of given inputs of the problem. Our results seamlessly unify linear uniform and nonlinear non-uniform problems and lead to an iteration algorithm for uniformly approximating the desired deflection.

This paper deals with an analytical approach to predict the nonlinear buckling behavior of functionally graded graphene-reinforced composite laminated cylindrical shells under axial compressive load surrounded by Pasternak’s elastic foundation in a thermal environment. Piece-wise functionally graded graphene-reinforced, composite layers are sorted with different types of graphene distribution. The governing equations are established by using Donnell’s shell theory with von Kármán nonlinearity terms and three-term solution of deflection is chosen for modeling the uniform deflection of pre-buckling state, linear and nonlinear deflection of post-buckling state. Galerkin method is applied to determine the critical axial compressive buckling load expression, post-buckling load-deflection and load-end shortening relations of the shell. The effects of environment temperature, foundation, geometrical properties, and graphene distribution on buckling behavior of shell, are numerically evaluated.

In this study, the dispersion of an infinite shear beam with a lumped mass connected at periodic distances and resting on an elastic foundation was examined. The effect of periodicity in the finite region of the lumped masses on wave propagation was investigated through a one-dimensional model. The dispersion relationship for Bragg scattering, which consists of one-dimensional periodic lumped masses, was derived using the transfer matrix method. Subsequently, to evaluate the effect of parameters such as the magnitude of the lumped mass and foundation stiffness on the dynamic response of the shear beam, several simulations were performed. The band frequency characteristics of the shear beam are demonstrated with respect to the variations in stiffness and mass. Using the wave-based approach, the effect of periodic masses on wave propagation in a finite region of an infinite beam was revealed. Periodic masses have been shown to have a positive effect on the displacement amplitude; in other words, a lumped mass barrier is effective in providing wave attenuation.

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

Under thermal environment and axial forces, the dynamic instability of functionally graded graphene platelet (GPLs)-reinforced porous beams on an elastic foundation is investigated. Three modes of porosity distributions and GPL patterns are considered. The governing equations are given by the Hamilton principle. On the basis of the differential quadrature method (DQM), the governing equations are changed into Mathieu–Hill equations, and the main unstable regions of the porous composite beams are studied by the Bolotin method. Thermal buckling and thermo-mechanical vibration problems are also studied. The effects of porosity coefficients and GPL weight fraction, dispersion pattern, initial thermal loading, slenderness ratio, geometry and size, boundary conditions, and foundation stiffness are discussed. The conclusions show that an elastic foundation has an obvious enhancement effect on thermal buckling, free vibration, and dynamic instability, which improves the stiffness of the beam.

This paper provides a general formularization of the nonlocal Euler–Bernoulli nanobeam model for a bending examination of the symmetric and asymmetric cross-sectional area of a nanobeam resting over two linear elastic foundations under the effects of different forces, such as axial and shear forces, by considering various boundary conditions’ effects. The governing formulations are determined numerically by the Generalized Differential Quadrature Method (GDQM). A deep search is used to analyze parameters—such as the nonlocal (scaling effect) parameter, nonuniformity of area, the presence of two linear elastic foundations (Winkler–Pasternak elastic foundations), axial force, and the distributed load on the nanobeam’s deflection—with three different types of supports. The significant deductions can be abbreviated as follows: It was found that the nondimensional deflection of the nanobeam was fine while decreasing the scaling effect parameter of the nanobeams. Moreover, when the nanobeam is not resting on any elastic foundations, the nondimensional deflection increases when increasing the scaling effect parameter. Conversely, when the nanobeam is resting on an elastic foundation, the nondimensional deflection of the nanobeam decreases as the scaling effect parameter is increased. In addition, when the cross-sectional area of the nanobeam varies parabolically, the nondimensional deflection of the nonuniform nanobeam decreases in comparison to when the cross-sectional area varies linearly.

Nonlinear buckling and postbuckling analysis of functionally graded graphene-reinforced composite (FG-GRC) laminated toroidal shell segments subjected to external pressure surrounded by elastic foundations and exposed to thermal environment are presented in this article. Governing equations for toroidal shell segments are based on the Donnell shell theory taking into account geometrical nonlinearity term in von Kármán sense with shell–foundation interaction modeled by Pasternak's elastic foundation. Three-term solution form of deflection and stress function are chosen, and Galerkin method is applied to obtain the nonlinear load–deflection relation. Numerical investigations show the effects of graphene volume fraction, graphene distribution types, geometrical properties, elastic foundation, and thermal environments on the linear and nonlinear buckling and postbuckling behaviors of FG-GRC laminated toroidal shell segments. [ABSTRACT FROM AUTHOR]

The paper focuses on the nonlinear vibration of functionally graded graphene nanoplatelet reinforced composite doubly curved shallow shells resting on elastic foundations. The graphene nanoplatelet reinforced composites are assumed to be distributed uniformly and functionally graded through the thickness. The material properties are assumed to be temperature-dependent and are estimated through the Halpin–Tsai micromechanical model, while the Poisson's ratio, density mass, and thermal expansion are implemented by the rule of mixtures. The mathematical formulation is developed based on the classical shell theory and Von Karman-Donnell geometrical nonlinearity assumption. The dynamical responses of a simply supported functionally graded-graphene nanoplatelet reinforced composite doubly curved shallow shells are obtained by employing the Airy's stress function and the Galerkin's method. The responses of nonlinear vibration as time history, frequency-amplitude curve, phase plane graphs, and Poincare maps are carried out in this paper. In addition, the effects of the environment, graphene nanoplatelets weight fraction, graphene nanoplatelets distribution patterns, and thickness-to-length ratio are scrutinized. The obtained results are also compared and validated with those of other studies. [ABSTRACT FROM AUTHOR]

This work deals with the wave dispersion characteristics of nanobeams considering thickness stretching effect under the influence of hygrothermal environment. The effect of initial stress is also analyzed. Since the displacements in the thickness direction are approximated as a parabolic function satisfying the upper and lower stress-free boundary conditions, this theory can account for both transverse-shear deformation and stretching effects. To capture the size-dependent effects, a high-order nonlocal strain gradient elasticity theory involving three small-scale coefficients is used. The governing equations of wave motion are obtained through Hamiltonian principle, and an analytic technique is applied for finding the wave frequencies as well as phase velocities. The influences of small-scale parameters, initial stress, hygrothermal environment, elastic foundation, and wave number on the wave characteristics of nanobeams are investigated. [ABSTRACT FROM AUTHOR]

This article proposes a finite element method (FEM) based on a quasi-3D nonlocal theory to study the free vibration of functionally graded material (FGM) nanoplates lying on the elastic foundation (EF) in the thermal environment. By applying Hamilton's principle, the governing equations of FGM nanoplates on the EF are obtained. Using the FEM helps solve many complicated problems that analytical solution (AS) cannot be performed yet, such as complex structures, asymmetric problems, variable thickness, etc. The numerical results of this work are compared with those of other published researches to verify accuracy and reliability. In addition, the effects of geometrical parameters, material properties such as the thickness, material exponents, nonlocal coefficients, elastic foundation stiffness, boundary conditions (BCs), and temperature on the free vibration of nanoplates are comprehensively investigated.

This study investigates the backflow of a Newtonian fluid in a two-dimensional flat-walled fracture with Navier slip boundary conditions. The fracture has a uniform aperture and two rigid pre-strained plates as walls; their elastic deformations are described by the Winkler model. Under the lubrication assumption, the governing nonlinear ordinary differential equation and the time-dependent velocity profile are derived; in turn, this yields the time and space evolution of the pressure distribution inside the fracture, numerically. In addition, the condition when the external pressure becomes zero, is discussed, and a parametric study is performed to highlight the influence of the slip length.

This study presents a hyperbolic shear deformation theory for free vibration of functionally graded plates on elastic foundations. The field of displacements is chosen based on the assumptions that axial and transverse displacements consist of components due to bending and shear. The components of the axial shear displacements give rise to the parabolic variation in the shear strain through the thickness, such that the shear stresses vanish on the plate boundaries. Therefore, the shear correction factor is not necessary. The material properties of the functionally graded plate are assumed to vary through the thickness according to the power law of the volume fraction of the constituents. The elastic foundation is modeled as a Pasternak foundation. The equations of motion are derived using Hamilton’s principle. The analytical solutions were established from Navier’s approach, and the results obtained are found to be in good agreement with the solutions of three-dimensional elasticity and with the solutions of the various plate theories. The effects of the power law index, the thickness ratio, and the foundation parameters on the natural frequency of the plates were also evaluated.

Despite the popularity of a completely free beam resting on a tensionless foundation in the construction industry, the existing bending analysis solutions are limited to certain types of loads (mostly point and uniformly distributed loads); these are also quite complex for practicing engineers to handle. To overcome the associated complexity, a simple iterative procedure is developed in this study, which uses the Ritz method for the bending analysis of a free–free beam on a tensionless foundation subjected to a patched load. The Ritz method formulation is first presented with polynomials being used to approximate the beam deflection with unknown constants to be determined through minimization of the potential energy. To account for the tensionless action, the subgrade reaction is set to zero when the deflection is negative. The non-zero subgrade reaction zone is defined by αlL/2

Buckling and postbuckling behavior of thin composite cylindrical shells reinforced by carbon nanotubes (CNTs), surrounded by elastic media and exposed to uniform temperature rise, are investigated in this article. CNTs are reinforced into isotropic matrix phase through uniform distribution or functionally graded distributions across the thickness direction. Material properties are assumed to be temperature dependent, and effective elastic moduli of CNT-reinforced composite are determined according to extended rule of mixture. Formulations are based on the classical thin shell theory taking Von Karman–Donnell nonlinearity, surrounding elastic media and elastic constraints of boundary edges into consideration. Multi-term solutions of deflection and stress function are assumed to satisfy simply supported boundary condition, and Galerkin method is applied to obtain nonlinear relation of thermal load and deflection. An iteration algorithm is used to determine buckling temperatures and postbuckling paths. Numerical examples are given to analyze the effects of volume fraction and distribution type of CNTs, geometrical parameters, degree of tangential edge constraint, buckling mode, and surrounding elastic media on the buckling temperatures and postbuckling strength of thermally loaded nanocomposite cylindrical shells. [ABSTRACT FROM AUTHOR]

This paper proposes a finite element method (FEM) based on a nonlocal theory for analyzing the free vibration of the functionally graded porous (FGP) nano-plate with different shapes lying on the elastic foundation (EF). The FGP materials with two-parameter are the power-law index (k) and the porosity volume fraction (J) in two cases of even and uneven porosity. The EF includes Winkler stiffness (k1) and Pasternak stiffness (k2). Some numerical results in our work are compared with other published to verify accuracy and reliability. Moreover, the influence of geometric parameters, materials on the free vibration of the FGP nano-plates resting on the EF is comprehensively investigated. [ABSTRACT FROM AUTHOR]

• Comprehensive vibrational analysis of anti-symmetric laminated composite plates. • Vibrational analysis includes an elastic foundation and thermal effects. • HSDT is applied to estimate the effects of the higher-order transverse shear. • Effects of damping, elastic, aspect and slenderness ratios are discussed in detail. • Different anti-symmetric laminates and boundary conditions are considered. In this work, a comprehensive vibrational behavior analysis is performed on anti-symmetric laminated composite plates resting on visco-elastic foundations undergoing thermal effects. Here, the governing equations of motion are developed through Hamilton's principle and Reddy's plate theory as higher-order shear deformation theory (HSDT) is employed to capture high accuracy. Also, the generalized differential quadrature method (GDQM) is used to predict the vibration response and the natural frequencies. The effects of temperature change, Winkler-Pasternak and damping coefficients for the elastic foundation, the elastic ratio, the arrangement of different anti-symmetric laminates, and the aspect and slenderness ratios are observed and discussed in detail. The results are extracted for fully clamped boundary conditions and the effects of other boundary conditions are also illustrated. [ABSTRACT FROM AUTHOR]

Double-cantilever beams (DCBs) are widely used to study mode-I fracture behavior and to measure mode-I fracture toughness under quasi-static loads. Recently, the authors have developed analytical solutions for DCBs under dynamic loads with consideration of structural vibration and wave propagation. There are two methods of beam-theory-based data reduction to determine the energy release rate: (i) using an effective built-in boundary condition at the crack tip, and (ii) employing an elastic foundation to model the uncracked interface of the DCB. In this letter, analytical corrections for a crack-tip rotation of DCBs under quasi-static and dynamic loads are presented, afforded by combining both these data-reduction methods and the authors' recent analytical solutions for each. Convenient and easy-to-use analytical corrections for DCB tests are obtained, which avoid the complexity and difficulty of the elastic foundation approach, and the need for multiple experimental measurements of DCB compliance and crack length. The corrections are, to the best of the authors' knowledge, completely new. Verification cases based on numerical simulation are presented to demonstrate the utility of the corrections. [ABSTRACT FROM AUTHOR]

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A new static–dynamic equivalence approach for beam bending is developed to determine the critical stability load of a vibrating beam resting on an elastic foundation. It is validated that the critical stability load corresponding to an arbitrary natural frequency parameter can be obtained via both the vibration and the static analysis approaches. By introducing a new elastic foundation stiffness parameter, the natural frequency parameter can be substituted, thus allowing a new formulation to solve the critical stability that corresponds to the zero or negative elastic foundation constant. In the vibration analysis, the governing equation of motion is solved by a harmonic response assumption. In this case, the critical stability loads can be determined by solving the determinant of the dynamic system under different boundary conditions. For zero or negative elastic foundation stiffness, a beam acted by a specific point load on such an elastic foundation can yield an infinite deflection. Under such circumstances, each real natural frequency has an equivalent relationship with the corresponding negative elastic stiffness. Consequently, using this equivalent system, directly solving of the transcendental equation in the dynamic frequency analysis can be avoided. Some numerical examples are presented and it is demonstrated that highly accurate numerical critical stability solutions can be derived by this equivalent static bending approach. The results are validated by comparing with the classical beam buckling solutions. In conclusion, a new theoretical model with analytical solution procedure is put forward and it yields highly convergent numerical solutions that compare well with classical analytical solutions. [ABSTRACT FROM AUTHOR]

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange's equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams' deflection results under a number of moving loads. [ABSTRACT FROM AUTHOR]

This work presents the effect of the elastic foundation and the viscoelastic interface on the dynamic behavior of laminated magneto-electro-elastic rectangular plates with simply supported boundary conditions using the state space method in Laplace domain. The Kelvin-Voigt model is used to take into accounted the viscoelastic interface effects in this domain. The final solution is transferred to the time domain by the Fourier inversion method. The dynamic responses of 3D displacements, stresses, and electric and magnetic displacements are analyzed with respect to the thickness direction and the orthotropic behavior under harmonic stress. A variant of the numerical tests shown the effect of the Winkler-Pasternak elastic foundation on a magneto-electro-elastic rectangular plates dynamic behavior and may contribute to optimize the design and the manufacturing of these materials. [ABSTRACT FROM AUTHOR]