Pasternak-type Elastic Foundation Research Articles

Large Deflection Geometrically Nonlinear Bending of Porous Nanocomposite Cylindrical Panels on Elastic Foundation

Large deflection nonlinear bending of functionally graded (FG) porous cylindrical panels reinforced with graphene platelets (GPLs) on a Pasternak-type elastic foundation is examined by developing a reliable and effective 2D meshfree-based nonlinear numerical method. The large displacement field is express by the first-order shear deformation theory (FSDT) and the von Kármán nonlinearity, and approximated by 2D natural element method (NEM) in conjunction with the stabilized MITC3+ shell concept and the shell surface–rectangular grid geometry transformation. The nonlinear simultaneous equations are solved by a load incremental Newton–Raphson scheme. The developed nonlinear numerical method is justified from by comparing with the reference solutions, and the load–deflection and bending moment of FG-GPLRC porous cylindrical panels on elastic foundation are scrutinizingly examined. Four different symmetric GPL distribution patters (except for FG-Λ) and three different symmetric porosity distributions are considered and their combined effects on the nonlinear bending behavior are investigated, as well as the effects of foundation stiffness and GPL amount. Also, the results are compared with those of FG CNT-reinforced porous cylindrical panels.

Mathematical modeling and solution of nonlinear vibration problem of laminated plates with CNT originating layers interacting with two-parameter elastic foundation

Generalizing the first-order shear deformation plate theory (FOPT) proposed by Ambartsumyan (Theory of anisotropic plates, Nauka, Moscow, 1967 (in Russian)) to the heterogeneous laminated nanocomposite plates and the nonlinear vibration problem is analytically solved taking into account an elastic medium in this study for the first time. The Pasternak-type elastic foundation model (PT-EF) is used as the elastic medium model. After creating the mathematical models of laminated rectangular plates with CNT originating layers on the PT-EF, the large amplitude stress–strain relationships and motion equations are derived in the form of nonlinear partial differential equations (PDEs) within FOPT. Then, by applying Galerkin's method to the derived equations, it is reduced to a nonlinear ordinary differential equation (NL-ODE) containing the second- and third-order nonlinear terms of the deflection function for laminated rectangular plates composed of nanocomposite layers. The NL-ODE is solved by the semi-inverse method, and the nonlinear frequency–amplitude relationship for the laminated plates consisting of CNT originating layers resting on the PT-EF is established within FOPT for the first time. From these relations, similar relations can be obtained particularly for the unconstrained laminated and monolayer CNT patterns plates. After comparing the accuracy of the obtained formulas with the reliable results in the literature, comprehensive numerical analyses are performed.

Static buckling analysis and geometrical optimization of magneto-electro-elastic sandwich plate with auxetic honeycomb core

The nonlinear static buckling analysis of magneto-electro-elastic sandwich plate on Pasternak-type elastic foundations subjected to the mechanical, thermal, electric and magnetic loadings is presented in this paper. The sandwich plate is composed of an auxetic honeycomb core with negative Poisson’s ratio and two face sheets made of magneto-electro-elastic​ material. The system basic equations are derived based on the Reddy’s higher order shear deformation plate theory taking into account the effect of von Kármán the kinematic nonlinearity and initial imperfection. The form of possible solutions and electric, magnetic potentials are chosen as trigonometric functions based on two cases of boundary conditions. The relationship between axial compressive loading and dimensionless deflection amplitude is determined by using the Galerkin method. For optimization problem, the Bees algorithm is applied to obtain the maximum value of critical buckling load of the sandwich plate which depends on five geometrical and material parameters. The effects of elastic foundations, temperature increment, geometrical parameters and electric and magnetic potentials on the stability characteristics are investigated in numerical results. The accuracy and reliability of present approach is confirmed by comparisons with the existing results in the literature.

Vibration analysis of auxetic laminated plate with magneto-electro-elastic face sheets subjected to blast loading

Mathematic modeling and analytical approach are presented for nonlinear vibration analysis of laminated plate with auxetic honeycomb core and magneto-electro-elastic face sheets supported by Pasternak-type elastic foundations. The laminated plate is subjected to the simultaneous action of blast, thermal, electric and magnetic loadings. The mechanical properties of auxetic core including negative Poisson’s ratio are assumed to depend on the geometrical parameters of the unit cells and elastic modulus of original material. The volume fraction of two components of each magneto-electro-elastic face sheet is chosen equally. The nonlinear motion equations and the geometrical compatibility equation are established by using Reddy’s higher order shear deformation plate theory taking into account the coupling between elastic, electric and magnetic fields. The analytical vibration solutions for the auxetic laminated plate can be obtained by using Galerkin and fourth-order Runge – Kutta methods. The numerical results are conducted to investigate the effect of geometrical and material parameters, elastic foundations, temperature increment, magnetic and electric potentials on the vibration characteristics of the auxetic laminated plate.

Nonlinear thermal dynamic buckling and global optimization of smart sandwich plate with porous homogeneous core and carbon nanotube reinforced nanocomposite layers

The analytical investigation for the nonlinear thermal dynamic buckling of smart sandwich plate subjected to mechanical, thermal and electric loadings is presented. The sandwich plate is composed of a porous homogeneous core, two carbon nanotube reinforced composite (CNTRC) layers and two piezoelectric face sheets. Basic equations are derived based on the Reddy's higher order shear deformation plate theory and Hamilton's principle in which the initial imperfection and Pasternak-type elastic foundations are included. The external pressure is assumed to be uniformly distributed on the surface of the sandwich plate and depend on time according to the linear functions. The nonlinear dynamic response, the frequency – amplitude relation are obtained by using the Galerkin and Runge – Kutta methods and the critical dynamic buckling load is determined by using Budiansky – Roth criterion. Bees Algorithm is used to determine the maximum value of natural frequency of smart sandwich plate and the corresponding optimum values of geometrical and material parameters. The effects of geometrical parameters, CNT volume fraction, elastic foundations, temperature increment, initial imperfection and porosity coefficient on the nonlinear vibration and dynamic buckling of the smart sandwich plate are considered specifically. The numerical results are also compared with existing results using different theories.

Nonlocal integral static problems of nanobeams resting on an elastic foundation

Eringen’s nonlocal theory has been widely used for the study of micro- and nano-structures exhibiting size effect phenomena. Eringen’s nonlocal differential constitutive equation has been employed to structural engineering applications, and a number of inconsistencies and paradoxes has been raised. This work centers on exploring static engineering benchmark problems of a nanobeam resting on a Pasternak-type elastic foundation by means of the nonlocal integral elasticity for the first time. The modified kernel’s model and the two phase stress model leading to well-posed problems are used. The static responses of the integral models show to have a flexible behavior in comparison with those of the classic and the nonlocal differential models for all the investigated problems. Neither paradoxes nor inconsistencies are raised for the integral models. The modified kernel highlights the model’s robustness from a physical point of view. The conclusions are promising to spur the applications of nanomaterials, nanocomposites and biomaterials.

Improvement of the low-frequency sound insulation of the poroelastic aerospace constructions considering Pasternak elastic foundation

This approach proposes a strategy based on modeling poroelastic doubly curved composite shells on a Pasternak-type Elastic Foundation (PEF). Likewise, a general formulation is developed considering Hamilton's principle to extract dynamic equations based on the shear deformation shallow shell theory (SDSST). Biot's study is also used to analyze wave propagation through a porous core. According to a finite vibroacoustic problem, it is essential to either excite the construction via an acoustic source or consider enough modes. Therefore, in addition to the formulation of the acoustic pressures based on the modal infinite double Fourier series, some factors are extended as a series harmonic solution. Before verification of the outcomes, some configurations with their convergence algorithm versus dimension and frequency are plotted. The results represent the effect of modeling poroelastic doubly curved composite shells subjected to a PEF on acoustic characteristics regarding each of Winkler spring and shear layer stiffness. Additionally, the transverse vibration of the construction is explored for the angle of incidence.

Nonlinear analysis of laminated FG-GPLRC beams resting on an elastic foundation based on the two-phase stress-driven nonlocal model

In this paper, a nonlinear formulation for beam-type structures is presented within the framework of two-phase stress-driven (SD) nonlocal theory. Various boundary conditions are considered for the beams, and it is assumed that they are on the Winkler- and Pasternak-type elastic foundations. It is considered that the beams are made of laminated functionally graded-graphene platelet-reinforced composite (FG-GPLRC) whose properties are estimated by means of the Halpin–Tsai model. The Euler–Bernoulli beam theory is also used for the modeling. The governing equations are derived based on the integral form of SD nonlocal theory using a variational approach considering geometrical nonlinearity. The equations of the SD model in differential form in conjunction with associated constitutive boundary conditions are also obtained. Moreover, a numerical approach based upon the generalized differential quadrature (GDQ) method is developed for the solution of the nonlinear bending problem. The influences of volume fraction/distribution pattern of GPLs, nonlocality, elastic foundation, and geometrical parameters on the bending response of beams under different end conditions are investigated. Furthermore, comparisons are given between the linear and nonlinear results.

The effect of considering Pasternak elastic foundation on acoustic insulation of the finite doubly curved composite structures

This approach focuses on the acoustic characteristic of a simply supported doubly curved composite shell subjected to the Pasternak-type elastic foundation. To carry out the sound analysis of a finite shell, displacements and rotation terms and acoustic pressures are developed based on the infinite longitudinal and transversal modes. According to Hamilton's principle, the dynamic equations of the structure equipped with an elastic foundation are extracted. Subsequently, the construction is stimulated using an acoustic wave. Considering infinite modes, the necessity of terminating this process by applying a large number of modes is concerned. Therefore, in addition to the design of the convergence algorithm, some configurations against variations of dimensions and frequencies are proposed. Before presenting the numerical outcomes, the accuracy of the formulation is checked by either natural frequencies or sound transmission spectra. It is realized that although Winkler spring stiffness is impressive to improve the noise property at the low-frequency domain, the positive effects of Shear layer one are specified in the whole region. The outcomes also contain some 3D new shapes for variations of the Winkler spring and Shear layer stiffness.

An analytical approach for nonlinear thermo-electro-elastic forced vibration of piezoelectric penta – Graphene plates

This paper deals with the nonlinear forced vibration of imperfect penta – graphene plates integrated with piezoelectric actuator layers. The plate is subjected to combination of mechanical, thermal, and electrical loadings. Based on the first order shear deformation plate theory, the governing equations are established taken into account the effect of the von Kármán type of geometrical nonlinearity, the Pasternak type elastic foundations, the damping and the piezoelectric – thermal effects. Four edges of the hybrid plate are assumed to be simply supported and immovable in the in-plane directions. The solution forms that satisfy the boundary conditions are assumed to be trigonometric. The closed form expressions of natural frequency, the frequency ratio – amplitude and the deflection amplitude – time curves are obtained by using the Galerkin and Runge – Kutta methods. The numerical results show positive effects of elastic foundations, negative effect of temperature increment and initial imperfection, considerable effect of geometrical parameters as well as small effect of applied voltage on the nonlinear forced vibration of piezoelectric penta – graphene plate.

Analytical solutions for nonlinear magneto-electro-elastic vibration of smart sandwich plate with carbon nanotube reinforced nanocomposite core in hygrothermal environment

This paper presents an analytical approach on the nonlinear magneto-electro-elastic vibration of smart sandwich plate. The sandwich plate consists of a carbon nanotube reinforced nanocomposite (CNTRC) core integrated with two magneto-electro-elastic face sheets. For core layer, three types of carbon nanotube (CNT) distribution such as FG-O, FG-V, FG-X are considered while the volume fraction of BaTiO3 − CoFe2O4 in each face sheet is chosen to be 0.5. It is assumed that the smart sandwich plate is rested on Pasternak-type elastic foundations and subjected to the combination of external pressure, thermal, electric and magnetic loads. The coupled constitutive relations are derived based on the Hamilton's principle in which the kinematic nonlinearity is defined using Reddy's higher order shear deformation theory. The analytical solutions which satisfy the boundary conditions are assumed to have the trigonometric form. The Galerkin method is used to obtain the closed form expressions of natural frequency, the relation between the frequency ratio and dimensionless amplitude and the dynamic response of the sandwich plate. The numerical results are conducted to illustrate the effect of geometrical parameters, CNT volume fraction, temperature and moisture increment, electric and magnetic potentials on the nonlinear vibration of smart sandwich plate. The reliability of present results is evaluated by comparing with the previous results based on different approach.

Eigenfrequencies of microtubules embedded in the cytoplasm by means of the nonlocal integral elasticity

A biologically microscopic system presenting a highly scientific interest is the microtubule (MT). Our research endeavor revolves around the eigenfrequencies’ analysis of an MT embedded in the cytoplasm of the cell by means of the nonlocal integral elasticity for the first time. The MT is simulated as a beam and the cytoplasm as a Pasternak-type elastic foundation, respectively. The responses of the nonlocal integral stress models show to have a softening behavior in comparison with that of the classic model. Unlike the nonlocal differential model, no paradoxes and inconsistencies are raised for the nonlocal integral models. Our research conclusions are a hopeful sign for the applications of biomaterials and bioengineering structures.

Buckling and Free Vibration Analysis of Fiber Metal-laminated Plates Resting on Partial Elastic Foundation

This research presents, buckling and free vibration analysis of fiber metal-laminated (FML) plates on a total and partial elastic foundation using the generalized differential quadrature method (GDQM). The partial foundation consists of multi-section Winkler and Pasternak type elastic foundation. Taking into consideration the first-order shear deformation theory (FSDT), FML plate is modeled and its equations of motion and boundary conditions are derived using Hamilton's principle. The formulations include Heaviside function effects due to the nonhomogeneous foundation. The novelty of this study is considering the effects of partial foundation and in-plane loading, in addition to considering the various boundary conditions of FML plate. A computer program is written using the present formulation for calculating the natural frequencies and buckling loadings of composite plates without contacting with elastic foundation and composite plates resting on partial foundations. The validation is done by comparison of continuous element model with available results in the literature. The results show that the constant of total or partial spring, elastic foundation parameter, thickness ratio, frequency mode number and boundary conditions play an important role on the critical buckling load and natural frequency of the FML plate resting on partial foundation under in-plane force.

Nonlinear post-buckling and vibration of 2D penta-graphene composite plates

The newly developed penta-graphene is a two-dimensional (2D) carbon allotrope with promising mechanical properties. This paper investigates the nonlinear post-buckling and vibration of imperfect three-dimensional penta-graphene composite plates resting on elastic foundations and subjected to uniform external pressure and axial compressive load. The elastic constants of the single-layer penta-graphene are fully determined by the density functional theory by fitting the equation of strain energy to the density functional theory energy. Specifically, the elastic constant $$C_{66}$$ which has not been considered by other authors is also determined. The motion and compatibility equations are derived based on the classical plate theory taking into account von Karman geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundations. For nonlinear post-buckling, the Bubnov–Galerkin method is applied to obtain the load–deflection amplitude curves while the Runge–Kutta method and harmonic balance method are used to obtain the deflection amplitude–time curves and the amplitude–frequency curves for nonlinear vibration. Numerical results show the effects of geometrical parameters, initial imperfection and elastic foundations on the nonlinear post-buckling and vibration of the imperfect 2D penta-graphene plates. The present results are also compared to others to validate the accuracy of the applied method and approach.

Higher-Order Stability Analysis of Imperfect Laminated Piezo-Composite Plates on Elastic Foundations Under Electro-Thermo-Mechanical Loads

This article provides a fully analytical approach for nonlinear equilibrium path of rectangular sandwich plates. The core of structure is made of symmetric cross-ply laminated composite and the outer surfaces are piezoelectric actuators which perfectly bonded to inner core. The structure is subjected to electro-thermo-mechanical loads simultaneously. One side of plate is rested on Pasternak type elastic foundation. The equilibrium equations of plate are derived based on the higher-order shear deformation theory of Reddy taking into account initial geometrical imperfection, nonlinear strain-displacement relations of von-Karman, temperature dependent properties, and different types of boundary conditions. Some numerical examples are presented to verify the accuracy of the proposed formulation. The effects of various parameters such as voltage on actuators, elastic foundation, imperfection, and pre-load condition on the buckling and postbuckling behaviors are studied. As an important finding of current research, there may be exists bifurcation point for imperfect plates by applying voltage on actuators.

Nonlinear vibration of FGM moderately thick toroidal shell segment within the framework of Reddy’s third order-shear deformation shell theory

Nonlinear vibration and dynamic response of functionally graded moderately thick toroidal shell segments resting on Pasternak type elastic foundation are investigated in this paper. Functionally graded materials are made from ceramic and metal, and the volume fraction of constituents are assumed to vary through the thickness direction according to a power law function. Reddy’s third order shear deformation, von Karman nonlinearity, Airy stress function method and analytical solutions are used to derive the governing equations. Galerkin method is used to convert the governing equation into nonlinear differential equation, then the explicit expressions of natural frequencies and nonlinear frequency–amplitude relations are obtained. Using Runge–Kutta method, the nonlinear differential equation of motion is solved, and then nonlinear vibration and dynamic response of shells are analyzed. The effects of temperature, material and geometrical properties, and foundation parameters on nonlinear vibration and dynamic characteristics are investigated and discussed in detail.

Thermal buckling and postbuckling of edge-cracked functionally graded multilayer graphene nanocomposite beams on an elastic foundation

This paper investigates thermal buckling and postbuckling behaviors of functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer beams containing an open edge crack and resting on a Pasternak-type elastic foundation based on the first-order shear deformation beam theory including von Kármán geometric nonlinearity. The material properties of functionally graded GPL-reinforced composites (GPLRCs), which exhibit piece-wise variation along the thickness direction, are evaluated using micromechanics based models. The bending stiffness of the cracked section is estimated by the rotational spring model. The obtained nonlinear partial differential equations of equilibrium are discretized by the differential quadrature method, and then an iterative method is used to obtain the thermal buckling loads and postbuckling load-deflection curves. Detailed parametric studies are conducted to investigate the effects of crack length, GPL distribution pattern, GPL weight fraction, GPL length-to-width and length-to-thickness ratios, boundary conditions, and foundation stiffnesses on the thermal buckling loads and postbuckling response of the cracked GPLRC beams.

Free vibration and buckling analyses of edge-cracked functionally graded multilayer graphene nanoplatelet-reinforced composite beams resting on an elastic foundation

This study investigates the linear free vibration and elastic buckling behaviors of functionally graded (FG) multilayer graphene platelet (GPL)-reinforced composite beams containing a single edge crack and resting on a Pasternak-type elastic foundation. GPLs are randomly oriented, and their concentration varies layer-wisely through the beam thickness. A modified Halpin-Tsai model is employed to estimate Young’s modulus of the GPL-reinforced composite (GPLRC). The first-order shear deformation beam theory is applied in structural modeling. The bending stiffness of the cracked section is assumed to be equivalent to that of a massless rotational spring, which is related to the stress intensity factor (SIF) at the crack tip. SIFs of the cracked FG multilayer GPLRC beams are calculated by the finite element method. The differential quadrature method is used to discretize the equations of motion of the cracked beams. A parametric study is performed after a validation study for the present approach to illustrate the effects of crack location, crack length, boundary condition, GPL weight fraction, GPL distribution pattern, GPL size and geometry, and foundation stiffnesses on the fundamental frequencies and critical buckling loads.

Nonlinear dynamic response and vibration of shear deformable piezoelectric functionally graded truncated conical panel in thermal environments

The novelty of this study is using the analytical approach to investigate the nonlinear dynamic response and vibration of shear deformable functionally graded truncated conical panel with piezoelectric actuators, resting on Pasternak type elastic foundations in thermal environments. Material properties are graded in the thickness direction according to a simple power law distribution in terms of the fractions of constituents. The governing equations are derived based on the first order shear deformation shell theory with a von Karman – Donnell type of kinematic nonlinearity in which the Hamilton's principle is used to derive the equations of motion of piezoelectric functionally graded truncated conical panel. The those equations are solved by the Galerkin method and Runge – Kutta method to determine the nonlinear deflection amplitude – time curves and natural frequency of the functionally graded panel. In numerical results, the effects of applied actuator voltage, temperature increment, dimensional parameters, semi – vertex angle, material properties and elastic foundations on the nonlinear dynamic response and vibration of the piezoelectric functionally graded truncated conical panel are discussed in details. The approach are verified with the known results in the literature.

Postbuckling of pressure-loaded FG-GRC laminated cylindrical panels resting on elastic foundations in thermal environments

This paper investigates the postbuckling behavior of graphene-reinforced composite (GRC) laminated cylindrical panels under lateral pressure loading. The panels are placed in a thermal environment and are supported by a Pasternak-type elastic foundation. The GRC layers are arranged in a piece-wise functionally graded (FG) pattern in the panel thickness direction. The anisotropic and temperature dependent material properties of GRC layers are predicted using the extended Halpin-Tsai micromechanical model. The higher order shear deformation theory and the von Karman strain-displacement relationships are employed to obtain the governing differential equations of the panels which also include the effects of temperature variation and panel-foundation interaction. The postbuckling of a cylindrical panel involves the boundary layer effect and the effects of the initial geometric imperfections. The nonlinear prebuckling deformations of the panel are also considered in the solution process. The postbuckling equilibrium path for the perfect and geometrically imperfect GRC laminated cylindrical panels are obtained by applying a singular perturbation technique along with a two-step perturbation approach. It is observed that, in the present examples, the considered FG-GRC laminated cylindrical panels subjected to lateral pressure experience pre-buckling deformation and the postbuckling equilibrium paths of the panels are no longer of the bifurcation type.