Neutral Surface Concept Research Articles

Nonlocal quasi-3d vibration/ analysis of three-layer nanoplate surrounded by Orthotropic Visco-Pasternak foundation by considering surface effects and neutral surface concept

The present article deals with the application of refined plate theory and surface effects to investigate the nonlocal free vibration behavior of a functionally graded nano-plate composed of two piezoelectric face sheets resting on Orthotropic Visco-Pasternak. Surface and nonlocal effects are applied using the surface piezoelectricity and stress-strain gradient theories. Surface effects, refined quasi-3D higher-order theory, neutral surface position and in-plane mechanical preloads are considered to study piezoelectric sandwich nano-plate with a functionally graded core in this study for the first time, simultaneously. The size-dependent governing equations are derived based on 2D and 3D refined theory using Hamilton’s principle and different boundary conditions are studied and investigated according to the semi-analytical solution method. Comparing the results of the vibration behavior of 2D and quasi-3D models, local and non-local models, and also, examining different foundations simultaneously can be said as one of the innovations of the current research. Finally, a parametric study is presented to examine the effects of different parameters such as surface effects, small scale parameters, mechanical preloads, thickness stretching effects, neutral surface position, non-homogeneous coefficient, applied voltage, different foundation constants, boundary conditions, geometrical ratios and different theories in details. It is indicated that the inclusion of surface effects and thickness stretching effects leads to a significant influence on the vibration behavior of three-layered nano-scale plates. The presented results in this article may provide useful guidance for the design and development of the next generation of nano-devices.

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

This paper investigates the dynamic behavior of micro/nanobeams made of two-dimensional functionally graded porous material (2DFGPM) under accelerated, decelerated, and uniform moving harmonic load, using surface elasticity and modified couple stress theories. The key feature of this formulation is that it deals with a higher order shear deformation beam theory. The non-classical equilibrium equations are developed using Lagrange's equation and the concept of physical neutral surface. The equations of motion are derived using the same approach, accounting for the porosity effect and the modified power-law distribution of material properties. The trigonometric Ritz method is used with sinusoidal trial functions for the displacement field, and the Newmark method is applied to obtain the dynamical response of 2DFGPM nanobeams. The results are compared with previous studies, and the impact of critical parameters such as gradation indices, volume fraction ratio, pattern of porosity, velocity, frequency, and motion type of the applied force are explored. This study highlights the importance of considering the porosity effect, as neglecting it can lead to significant errors in the predicted results. Additionally, the study found that the accelerated and decelerated motions of the applied load have a greater impact on the dynamical deflection of 2DFGPM nanobeams than the uniform motion. The findings of this study can provide guidance for the optimal design of micro/nanobeams subjected to a moving force with multifunctional properties.

Size-dependent vibrations of bi-directional functionally graded porous beams under moving loads incorporating thickness effect

This article analyzes the forced and free vibrations of bi-directional functionally graded (BDFG) porous nanobeams under moving loads by considering the thickness effect based on the nonlocal strain gradient theory (NSGT). It is assumed that the material properties of the system are graded across the transverse and axial directions according to power and exponential laws, respectively. Various porosity models, including even, uneven, and asymmetric porosity distribution patterns, are considered to model the porosity effects. The dynamic equation of the system is derived by implementing the physical neutral surface concept. Analytical and numerical approaches are adopted to acquire the vibration response of the system. Mathematical closed-form expressions are provided for various dynamic phenomena, such as two types of cancelation, two types of resonance, maximum resonance, cancelation disappearance, and resonance disappearance. In addition, for the first time in this article, it is demonstrated that BDFG porous nanobeams under moving loads are prone to experience the double cancelation phenomenon. Several comparison studies with published data are conducted to validate the results. Also, comprehensive parametric studies are carried out to elucidate the impacts of system parameters such as thickness power index, axial gradient parameter, scale parameter ratio, axial force, porosity coefficient, and porosity distribution pattern on the dynamic magnification factor, critical load speed, and maximum free-response. The outcomes established that by considering the thickness effect, the stiffening behavior of the system is amplified depending on the slenderness ratio. Also, it is concluded that, among the considered porous models, the critical load speed for the system with an even porosity distribution is more dependent on the thickness power index.

Thermomechanical vibration and stability of effectively homogenized FGM beam with temperature-dependent shear correction factors

Thermomechanical characteristics in the Functionally Graded Materials (FGMs) have been investigated in depth for the precise modeling and analysis of structures in ultimate thermal environments. On this subject, the micro-mechanical interactions of particles in the mixtures of FGMs is necessary to consider the more accurate estimation of structural model. Therefore, the effective material properties are obtained by using homogenization technique based on Mori-Tanaka scheme (MTS) with temperature-dependent material properties. In this regard, present research studies detail temperature-dependent frequencies and thermal buckling behavior of beam model. Then, the physical neutral surface concept is adopted to formulate the decoupled set of stress–strain relations for the FGM models. Also, the First-order Shear Deformation Theory of Beam with rectangular cross-sectional shape including temperature-dependent shear correction factors is developed to improve the practical applications. Furthermore, the results compared with previous data, and the present model is studied in the broad range of temperatures. Specifically, the correlations of shear correction factors and neutral surface shifts are discussed for the models according to temperature and volume fractions. Finally, the thermal stability and the natural frequencies of vibration behaviors are discussed as a result of the mixture ratio for the materials. Furthermore, the mutual interactions of micro-mechanical behavior with the correction factors are especially investigated for thermomechanical analysis.

Temperature-dependent shear correction factor with heat transfer based on micromechanical properties for FGM plates

For a development of aerospace structures in high-temperature regions, Functionally Graded Materials (FGMs) have been used rather than conventional composites. In this regard, present work considers the FGM models in thermal environments, and crucial evaluation to supplement the heat conduction is investigated. To model the thermal–structural interactions, the First-order Shear Deformation Theory is adopted with temperature-dependent characteristics including thermo-micromechanical properties. And the material distributions through the thickness direction are then examined in two types such as the Power- and the Sigmoid-laws. Furthermore, the Mori–Tanaka method for homogenization technique is employed for the precise micromechanical interactions of the mixture particles. Moreover, the un-symmetry of material properties in the structure is considered with physical neutral surface concept. To ensure the validity of this study, the previous literatures are used to compare with the present numerical results. And then, shear correction factors are fully discussed using homogenization methods and the temperature-dependent material properties for micromechanical modeling. Additionally, the effect of heat conduction is studied in detail according to the Poisson’s ratio and Young’s modulus of the mixtures.

Nonlinear bending analysis of fgp plates under various boundary conditions using an analytical approach

In this paper, the nonlinear bending behavior of functionally graded porous (FGP) plates under uniformly distributed transverse loads is studied based on the neutral surface concept within the framework of first-order shear deformation theory (FSDT) including geometrical nonlinearity. The FGP materials with three porosity distribution patterns namely uniform, non-uniform symmetric and non-uniform asymmetric are considered. Using the stress function and Galerkin method, the analytical solutions are obtained to examine the load vs deflection and load vs bending moment curves for various edges boundary conditions. The present results are compared with reference solutions to show the accuracy and the effectiveness of the proposed approach. The effects of the geometric, material parameters, elastic foundations and in-plane constraints on the nonlinear bending behavior of FGP plates are studied in detail through numerical investigations.

Exact solution for nonlinear static behaviors of functionally graded beams with porosities resting on elastic foundation using neutral surface concept

In this paper, an exact solution for nonlinear static behaviors of functionally graded (FG) beams with porosities resting on the elastic foundation is presented. The FG material properties with porosities are assumed to vary along the thickness of the beam, and two types of porosity distributions are considered. Actually, the geometrical middle surface of the FG beam selected in computations is very popular in the literature. By contrast, in this study, the physical neutral surface of the beam is utilized. Based on the Timoshenko beam theory, von Kármán nonlinear assumption, together with neutral surface concept, the nonlinear governing equations of the FG beam resting on the elastic foundation are derived. By using the physical neutral surface, the nonlinear governing equations have simple forms and can be solved directly. The exact solution for the problem with all immovable and moveable boundary conditions is conducted in detail. Some numerical investigations to show the effects of boundary conditions, material properties, length-to-thickness ratio, elastic foundation coefficients and several types of applied load on nonlinear static bending behaviors of the beam are given.

Nonlinear bending and vibration analyses of FG nanobeams considering thermal effects

Nonlinear bending and nonlinear free vibration analysis are presented for FG nanobeams based on physical neutral surface concept and high-order shear deformation beam theory with a von Kármán-type equations and including thermal effects. The material properties are temperature-dependent and vary in the thickness direction. Nonlinear bending approximate solutions and free vibration solutions for present model with fixed supported boundary conditions are given out by a two-step perturbation method. Some comparisons are presented to valid the reliability of the present study. In numerical analysis, the effects of the volume fraction, nonlocal parameter, strain gradient parameter, and temperature changes on nonlinear bending and vibration are investigated.

Nonlinear analysis of thermal-mechanical coupling bending of FGP infinite length cylindrical panels based on PNS and NSGT

The present paper investigates the nonlinear static bending behavior of infinite length cylindrical panels made of functionally graded porous (FGP) materials. The shell with clamped edges is subjected to uniform temperature rise and transverse pressure loading. Thermomechanical properties of the shell with even distributed porosities are temperature-dependent and are graded along the thickness. The principle of virtual displacement and nonlocal strain gradient theory (NSGT) are employed to derive the equilibrium equations of the shallow shell resting on nonlinear elastic foundation. The concept of physical neutral surface (PNS) and higher-order shear deformation theory are also included into the formulation. Using the uncoupled thermoelasticity and Donnell kinematic assumptions, three differential equations of the shell under thermomechanical loading are established. The nonlinear system of governing equations is solved for the shell with infinite length which is clamped on both straight edges and free at other curved edges. The two-step perturbation technique and Galerkin procedure are employed to derive analytical solutions for the thermal, mechanical, and thermomechanical responses of cylindrical panels. The important parameters governing the bending behavior of shells under thermal-mechanical coupling load are identified and discussed. Parametric studies are given to show the influences of nonlocal/length scale parameter, porosity coefficient, foundation stiffness, geometrical parameter and functionally graded pattern. It is shown that the geometrical parameters have an important role on the bending behavior of cylindrical panels. Also, the temperature dependence of material properties results in higher deflection of the heated shells.

Nonlinear thermal buckling and postbuckling analysis of bidirectional functionally graded tapered microbeams based on Reddy beam theory

In the present study, the nonlinear thermal buckling, postbuckling, and snap-through phenomenon of higher-order shear deformable tapered bidirectional functionally graded (BDFG) microbeams are comprehensively investigated under different types of thermal loading, for the first time. The thermomechanical properties of the BDFG microbeam are assumed to be functions of temperature, axial, and thickness directions. Reddy’s parabolic shear deformation beam theory with the von Karman nonlinearity is employed to derive the variable coefficient governing nonlinear differential equations on the basis the physical neutral surface concept. Size-dependent effect is captured in the formulation employing the modified couple stress theory. The generalized differential quadrature method (GDQM) is used to discretize the motion equations considering different boundary conditions. The resulting system of nonlinear algebraic equations is solved iteratively using Newton’s method. Theoretical analysis and numerical results indicate that, depending on the shear deformation beam theory, boundary conditions, and the type of thermal load, the response of the BDFG microbeam may be of the bifurcation or snap-through buckling type of instability. Numerical parametric studies are conducted to explore the influences of thermal load type, material property gradient indexes, boundary conditions, material temperature dependency, taperness ratio, and microstructural length scale on critical thermal buckling load, thermal postbuckling, and equilibrium paths of the BDFG microbeam.

Thermally induced nonlinear dynamic analysis of temperature-dependent functionally graded flexoelectric nanobeams based on nonlocal simplified strain gradient elasticity theory

In this work, thermally induced dynamic behaviors of functionally graded flexoelectric nanobeams (FGFNs) are analyzed theoretically while considering the neutral surface concept and the von Karman nonlinearity induced by thermal environment. The temperature field is assumed to vary only in the thickness direction by solving a simple steady state heat transfer equation and to be constant in the plane of the beam. The temperature-dependent material properties of FGFNs are assumed to vary continuously throughout the thickness according to a power-law form. For such FGFNs, the nonlocal simplified strain gradient elasticity theory to capture the effect of size-dependent and, the higher order shear deformation beam theory to account for rotary inertia and transverse shear strains are adopted to formulate the governing equations and associated boundary conditions, which are solved by using a two-step perturbation method. In the numerical part, comparison study is also performed to verify the present theoretical model and the parametric analysis is systematically studied. It is also found that the thermally induced bending amplitude, nonlinear frequency and frequency ratio depend enormously on the material distribution profile, the flexoelectricity, the size-dependent effect and the imposed temperature field.

Static behaviour of functionally graded plates resting on elastic foundations using neutral surface concept

Static behaviour of functionally graded plates resting on elastic foundations using neutral surface concept

A refined of trigonometric shear deformation plate theory based on neutral surface position is proposed for static analysis of FGM plate.

In this study, we developed a new theory of hyperbolic shear deformation of functionally graded plates(FGPs) based on the position of the neutral surface to analyze displacements and shear stresses in the case of static bending behavior. The theory accounts for hyperbolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The neutral surface position for a functionally graded plate which its material properties vary in the thickness direction is determined. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Based on the present new hyperbolic shear deformation plate theory and the neutral surface concept, the governing equations of equilibrium are derived from the principle of virtual displacements. Numerical illustrations concern flexural behavior of FG plates with Metal–Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions.

Study on the vibration of functionally graded microcantilevers immersed in fluids under photothermal excitation

In this study, the dynamic response of FGM (Functionally graded materials) micro-cantilever immersed in fluids under high-frequency photothermal excitation was investigated theoretically. The temperature along the length of microcantilever can be obtained analytically by using Fourier heat conduction theory. The axial thermal stress varying along the thickness can be obtained by the temperature distribution. Using concept of physical neutral surface and thermal stress, photothermal driving force was obtained analytically by using thermoelastic theory. The hydrodynamic force was presented by means of Sader’s method. Based on the Euler-Bernoulli beam model, effective bending modulus and effective density of the FGM cantilever, dynamical deflection fields in fluids can be obtained analytically by using mode superposition method. Theoretical analysis showed that the influence of volume fraction in vacuum or air is more significant than in fluids, and the volume fraction has a less influence when the dimension of microcantilever get smaller. This study can be valuable to users and designers of FG microcantilever-based structures in MEMS/NEMS.

Large amplitude free vibrations of long FGM cylindrical panels on nonlinear elastic foundation based on physical neutral surface

In the current investigation, the small and large amplitude free vibration characteristics of long cylindrical panels made of FGMs is investigated using the higher order shear deformation shell theory and Donnell kinematic assumptions. The interaction of the panel with a nonlinear hardening/softening foundation in also included into the formulation. The properties of the panel are assumed to be graded across the thickness of the panel and assumed to be temperature dependent. Using the neutral surface concept, the three coupled motion equations of the shell are established and solved for the case of long panels which are immovable simply supported on straight edges. The solution method is based on the two step perturbation technique which results in a closed form expression for the frequencies of the panel as a function of the mid-point deflection. Numerical studies are first validated for the case of long FGM plates. Afterwards novel numerical results are given to show the effect of geometrical parameters, power law index, foundation stiffnesses and temperature elevation. As shown, temperature elevation results in the reduction in natural frequencies and enhancement of the frequency ratio of the panel. Also temperature dependency reduces the natural frequencies and frequency ratios of the panel.

Bending analysis of functionally graded beam with porosities resting on elastic foundation based on neutral surface position

In this paper, the Timoshenko beam theory is developed for bending analysis of functionally graded beams having porosities. Material properties are assumed to vary through the height of the beam according to a power law. Due to unsymmetrical material variation along the height of functionally graded beam, the neutral surface concept is proposed to remove the stretching and bending coupling effect to obtain an analytical solution. The equilibrium equations are derived using the principle of minimum total potential energy and the physical neutral surface concept. Navier-type analytical solution is obtained for functionally graded beam subjected to transverse load for simply supported boundary conditions. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions. The influences of material parameters (porosity distributions, porosity coefficient, and power-law index), span-to-depth ratio and foundation parameter are investigated through numerical results.
 Keywords: functionally graded beam; bending analysis; porosity; elastic foundation; bending; neutral surface.
 Received 10 December 2018, Revised 28 December 2018, Accepted 24 January 2019

On buckling of porous double-layered FG nanoplates in the Pasternak elastic foundation based on nonlocal strain gradient elasticity

In the present investigation, the buckling behaviours of porous double-layered functionally graded nanoplates in hygrothermal environment are presented for the first time. The nonlocal strain gradient theory with two material scale parameters is developed to examine buckling behaviour much accurately. Based on the new first order shear deformation theory the equations of equilibrium are obtained from the principle of minimum potential energy. To simplify the equations of equilibrium and removing the bending-extension coupling, the buckling behaviours of FG nanoplates are investigated based on physical neutral surface concept. The equations of equilibrium are solved for various boundary conditions using Galerkin's method. The obtained results are compared with the results available in the literature to valid the correctness of present solution method. The effects of nonlocal parameter, strain gradient parameter, porosity volume fraction, power-law index, temperature change, humidity change and boundary conditions on critical buckling load are presented.

Application of sinusoidal shear deformation theory and physical neutral surface to analysis of functionally graded piezoelectric plate

Abstract The concept of neutral surface for a Functionally Graded Piezoelectric (FGP) plate is developed in this paper. The electro-elastic analysis of a FGP plate resting on Winkler-Pasternak foundation is performed in the theoretical framework provided by a two-variable sinusoidal shear deformation theory, including the aforementioned concept of neutral surface. First, the location of neutral surface is defined with respect to the position of the middle surface and then the governing bending equations are derived using the principle of virtual work. An analytical method is presented to investigate the variation of the displacement and stress components in terms of the main parameters of the model, which are volume fraction exponent of the constituents, the electric potential and foundation parameters. The numerical results are validated through the comparison with available references. The numerical results prove that the value of transverse bending deflection is greater than the corresponding shear deflection. In addition, it can be observed that the volume fraction exponent has peculiar influence on the distribution of displacements and stresses.

Exact solutions for postbuckling of a graded porous beam

An exact, closed-form solution for the postbuckling responses of graded porous beams subjected to axially loading is obtained. It was assumed that the properties of the graded porous materials vary continuously through thickness of the beams, the equations governing the axial and transverse deformations are derived based on the classical beam theory and the physical neutral surface concept. The two equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformations. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for postbuckled deformation is obtained as a function of the applied load. The exact solutions explicitly describe the nonlinear equilibrium paths of the buckled beam and thus are able to provide insight into deformation problems. Based on the exact solutions obtained herein, the effects of various factors such as porosity distribution pattern, porosity coefficient and boundary conditions on postbuckling behavior of graded porous beams have been investigated.

Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method

In this paper, free vibration behavior of thin functionally graded rectangular plates is investigated by using the dynamic stiffness method (DSM). Classical plate theory along with the concept of physical neutral surface of the functionally graded plate is used to formulate the dynamic stiffness matrix. The dynamic stiffness matrix is finally solved by using the Wittrick-Williams algorithm to compute the natural frequencies. DSM frequencies are compared with those available in the literature. Some inaccurate published results are pointed out and possible reasons for these inaccuracies are discussed. Results for several plate parameters are given and the influence of these parameters on natural frequencies of the functionally graded plate is highlighted. The present study shows that the dynamic stiffness method provides very accurate results for vibration analysis of thin functionally graded plates and these results can be used as benchmark solutions for comparison purposes.