Static Bending Behavior Research Articles

A new model to study magnetic-electric fields effects on bending of nano-scale magneto-electro-elastic beams

In this paper, a new 2D formulation is developed for analyzing the effects of magnetic and electric fields on the static bending behavior of magneto-electro-elastic (MEE) nanobeams subjected to transverse loading conditions. The nonlocal elasticity theory is utilized to consider the small-scale effect on the behavior of nanobeams. In order to increase the accuracy of the model to obtain more realistic predictions, a general displacement field is considered, which take into account a general form for the distribution of normal and shear strains in the nanobeam and to consider a general form for the distribution of magnetic and electric fields in the MEE nanobeam. Using the principle of minimum total potential energy, the governing equations of the MEE nanobeam under external magnetic, electric, and mechanical loadings are determined. An analytical solution is developed to investigate the effects of magnetic and electric fields on the bending of the MEE nanobeam. According to the numerical results, it is shown that the present method can predict the deflection of magneto-electro-elastic nanobeams more accurately in comparison with predictions of the shear deformation theories available in the literature, due to considering all in-plane and out of plane strain components and considering a general (not predefined) form for the distribution of electric and magnetic field through the thickness of MEE nanobeam. A detailed numerical study is presented to examine the influence of the nonlocal parameter, magnetic potential, electric potential, transverse and axial loadings, and aspect ratio on the deflection of magneto-electro-elastic nanobeam. The results revealed for BaTiO3–CoFe2O4 nanobeam that the maximum deflection of MEE nanobeam increases by applying a negative external magnetic potential as well as a positive external electric potential. Due to the excellent accuracy of the presented formulation, it can be used in the analysis and design of MEE nanostructures and as a benchmark for the investigation of the accuracy of other numerical models.

Bending and Vibration Analysis of Flexoelectric Beam Structure on Linear Elastic Substrates.

With the development of micro-nanotechnology, smart electronic devices are being updated and developed, and more and more flexoelectric sensors, actuators, and energy harvesters attached to elastic substrates have attracted a surge of interest due to unique features at the nano-scale. In this paper, the static bending behavior and vibration characteristics of a flexoelectric beam structure based on a linear elastic substrate under a magnetic field environment are investigated. Based on the electrical Gibbs free energy density, the governing equations and boundary conditions of structures are derived by using the Euler–Bernoulli beam theory and the Hamilton’s variational principle. The expressions of the deflection and the induced electric potential of the beam structure are expressed analytically. The natural frequency of the beam under the open-circuit electrical conditions with surface electrodes (OCI) are obtained after further extending the solution. The results show that the flexoelectric effect, the linear elastic substrate, and the magnetic field have significant effects on the static bending and vibration behaviors of the flexoelectric beam which are beneficial for designing and developing flexoelectric devices with elastic substrates.

Bending and Wave Propagation Analysis of Magneto-Electro-Elastic Functionally Graded Porous Microbeams

In this paper, a microstructure-dependent magneto-electro-elastic functionally graded porous (MEEFGP) beam model is proposed using a variational approach. To account for the microstructure effect, the extended modified couple stress theory is incorporated in the new model. In addition, the porosity variation of the two-phase beam model through the thickness direction is also considered. The new developed model is verified in terms of its correctness with a FEM model. Based on the equations of motion and boundary conditions derived by Hamilton’s principle, the static bending and wave propagation behaviors of the new model are analytically determined. The results prove the existence of the microstructure effect and the magneto-electro-elastic multi-field coupling effect. There are significant differences between the new model and the classical model at the microscale. Moreover, the porosity also has an important influence on the mechanical properties of the new model. The results predicted by the new model can provide the theoretical basis for the design of microscale acoustic wave devices and micro-electro-mechanical systems.

Bending, buckling and vibration analysis of Bi-directional functionally graded Circular/Annular microplate based on MCST

The modified couple stress theory (MCST) is applied to analyze axisymmetric static bending, elastic buckling, and free vibration behaviors of circular/annular microplates made of bi-directionally functionally graded (BiFG) materials. The material properties of functionally graded (FG) microplates are assumed to vary along both through-the-thickness and through-the-radius directions. The differential governing equation and boundary conditions are derived through the Hamilton's principle, and expressed in nominal form with the introduced nominal variables. The generalized differential quadrature method is applied to discretize the differential governing equation and boundary conditions. The bending deflection can be obtained through solving a set of linear equations, while buckling loads and vibration frequencies can be determined through solving general eigenvalue problems. The current model is verified with those existed in literature. Selected numerical results are illustrated to indicate the effects of material length scale parameter, FG material indices and inner-to-outer radius ratio on the bending deflection, buckling loads and free vibration frequency of BiFG circular/annular microplates subjected to different boundary conditions.

A Proposed Method to Evaluate the Effect of Changing the Kerfing Parameters upon the Static Bending Behavior of Flexible Plywood Panels Cut by Laser

The purpose of this paper is to propose a method for testing the effect of varying the kerfing parameters, on the flexibilization of plywood panels for indoor applications. The evaluation was made by comparisons of the change in the MOE, MOR, and maximum deflection, between flexibilized and non-flexibilized specimens subjected to static bending tests based on EN 310. In order to prevent the problem of sliding from the supports occurring for the flexibilized specimens, the standard specimens were modified by adding a frame and subjecting to bending just their central rail. Framed specimens of poplar plywood, of 8 mm thick, were laser cut with a rigid central rail, taken as reference and with the flexible kerfed rail. These had lengths of the flexible area of 50, 70, 90, and 110 mm and with two dimensions of the kerfing pitch, 6 and 10 mm. Very good correlations were found for MOE, MOR, and maximum deflection with the length of the kerfed area, for both values of the kerf pitch, which proves the sensitivity of the proposed method to the changes in the input parameters. The method could further serve to mathematically model the flexibility of a kerfed plywood panel by selecting the appropriate input data.

A refined hyperbolic shear deformation theory for nonlinear bending and vibration isogeometric analysis of laminated composite plates

In this paper, a refined hyperbolic tangent higher order shear deformation theory is developed. Assuming that the shear function is parameter dependent, the parameters are determined by the inverse method. The refined theory is assessed by using the NURBS based isogeometric analysis for the geometric linear and nonlinear bending and free vibration problems of laminated composite plates. The nonlinearity of the plates is based on the von-Karman strain assumptions. Numerical examples show that the refined hyperbolic tangent shear deformation theory combined with IGA has high accuracy in both linear and geometric nonlinear analysis of laminated composite plates. The effects of plate length–thickness ratio, modulus ratio and stacking sequence on the static bending and free vibration behaviors of laminated plate are also discussed.

Influence of Graded Surface Decarburization of Automobile Forging Front Axle on the Bending Behavior Based on a Third-Order Shear Deformation Beam Theory

During the forging process of automobile front axle, the steel near the surface is often decarburized for a certain depth. The mechanical properties at the decarburization layer are graded and different from the inner area, influencing the bending behavior of axles under heavy loads. In this paper, the decarburized forging of front axle is regarded as a rectangular thick sandwich beam, composed of a homogeneous core and the functionally graded layer coated on both bottom and top surface. A Third-order Shear Deformation Theory (TSDT) is employed to investigate the static bending behaviors under two point−loads. The properties of sandwich FG material are represented with a piecewise power−law function, and the displacement field governing equations are derived through the virtual work principle. The Navier analytical method and numerical DQM procedures are employed to obtain the bending responses under simply supported boundary conditions, and the results are validated through the comparison with an example in the literature. Then, the transverse deflection, rotation, axial stress, and shear stress are studied in terms of different power−law exponents, decarburization depth, unsymmetrical decarburization depth, unbalance loading, and beam sectional dimension. The study reveals the influence of surface decarburization on the bending behavior of forged automobile front axles, and contributes to the optimization of structure design.

Spatially curved functionally graded Timoshenko microbeams: A numerical study using IGA

In this paper, we present numerical investigations on static bending and free vibration behaviors of spatially curved functionally graded (FG) microbeam using isogeometric analysis (IGA). The material properties change continuously along the thickness direction. The small scale effects are captured using the modified couple stress theory while microbeam kinematics are described by Timoshenko beam theory, and the spatially curved microbeam is represented by the Frenet–Serret bases. The geometries of curved microbeams are exactly represented with the non-uniform rational B-spline (NURBS) basis functions, and the same basis functions are applied to obtain displacement field approximation. Numerical examples are presented and computed results are then compared with reference methods to show the accuracy of the developed method. The effects of some factors including material gradient factor, material length scale parameter, and boundary conditions on the mechanical behaviors of spatially curved FG Timoshenko microbeams are investigated.

A unified size-dependent plate theory for static bending and free vibration analyses of micro- and nano-scale plates based on the consistent couple stress theory

Based on the consistent couple stress theory (CCST), the authors develop a unified formulation of various shear deformation plate theories for static bending and free vibration analyses of simply-supported, micro-/nano-scale plates embedded in an elastic medium. The CCST-based classical plate theory (CPT), first-order shear deformation plate theory (SDPT), and Reddy's refined SDPT, as well as the CCST-based sinusoidal, exponential, and hyperbolic SDPTs can be obtained by assigning specific through-thickness distributions of the shear deformations into the unified formulation, and they are thus included as special cases of the unified consistent couple stress plate theory. The unified CCST-based plate theory is applied to static bending and free vibration analyses of simply-supported, functionally graded microplates (FGMPs) and multilayered graphene sheets (MLGSs) embedded in an elastic medium. The material length scale parameter, aspect ratio, and shear deformation effects on the deformation, stress, and frequency parameters of FGMPs and MLGSs are investigated. It is shown that the solutions for the deformation, the in-plane stress, and the frequency parameters of the FGMPs/MLGSs obtained using the MCST and the CCST are almost identical to each other. The material length scale parameter effects on static bending and free vibration behaviors of the FGMP/MLGS are significant.

On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams

In this paper, a new magneto-electro-elastic functionally graded Timoshenko microbeam model is developed by using the variational formulation. The new model incorporates the extended modified couple stress theory in order to describe the microstructure effect. The power-law variation through the thickness direction of the two-phase microbeams is considered. By the direct application of the derived general formulation, the static bending and free vibration behavior of the newly developed functionally graded material microbeams are analytically determined. Parametric studies qualitatively demonstrate the microstructural effect as well as the magneto-electro-elastic multi-field coupling effect. The proposed model and its classic counterpart produce significant differences for thin graded magneto-electro-elastic Timoshenko microbeams. The thinner the microbeam is, the larger the difference becomes.

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.

Static bending analysis of functionally graded sandwich beams using a novel mixed beam element based on first-order shear deformation theory

In this paper, the static bending behavior of the functionally graded sandwich beams is investigated using a novel mixed beam element based on the first-order shear deformation theory. The proposed beam element consists of two nodes and three degrees of freedom per node as the traditional Timoshenko beam element. By introducing a special process, selective or reduced integration are not required to calculate the element stiffness matrix and nodal force vector. The efficiency and accuracy of the proposed element are verified via some comparison studies. Then the proposed element is applied to study the bending behavior of the functionally graded sandwich beams with various boundary conditions. The influences of some parameters such as the power-law index, the slender ratio and the boundary conditions are studied carefully. The comparison studies and numerical results present that the new beam element is compatible with the static bending analysis of the functionally graded sandwich beams with a very coarse mesh. Besides, the numerical results show that the stress concentration may occur on the separated surface of the core layer and two face sheets, which should be noticed in practical applications of the FG sandwich beams.

A quasi-three-dimensional isogeometric model for porous sandwich functionally graded plates reinforced with graphene nanoplatelets

The purpose of this study is to present a quasi-three-dimensional (quasi-3D) shear deformation theory for static bending and free vibration analyses of porous sandwich functionally graded (FG) plates with graphene nanoplatelets (GPLs) reinforcement. In addition, we propose a novel sandwich plate model with various outstanding features in terms of structural performance. The quasi-3D theory-based isogeometric analysis (IGA) in conjunction with refined plate theory (RPT) is first exploited to capture adequately the thickness stretching effect for porous sandwich FG plate structures reinforced with GPLs. The Non-Uniform Rational B-Splines (NURBS)-based IGA is employed in order to describe exactly the geometry models as well as approximate the unknown field with higher-order derivatives and continuity requirements while the RPT model includes only four essential variables. The sandwich FG plates consist of a core layer containing internal pores reinforced by GPLs and two functionally graded materials (FGMs) skin layers. Effective mechanical properties can be evaluated by employing the Halpin-Tsai model along with the rule of mixture. Various combinations of two porosity distributions and three GPL dispersions in the core layer are thoroughly investigated. Several numerical investigations are conducted to examine the effects of several key parameters on the static bending and free vibration behaviors of sandwich FG plate structures.

Formulation of a New Mixed Four-Node Quadrilateral Element for Static Bending Analysis of Variable Thickness Functionally Graded Material Plates

A new mixed four-node quadrilateral element (MiQ4) is established in this paper to investigate functionally graded material (FGM) plates with variable thickness. The proposed element is developed based on the first-order shear deformation and mixed finite element technique, so the new element does not need any selective or reduced numerical integration. Numerous basic tests have been carried out to demonstrate the accuracy and convergence of the proposed element. Besides, the numerical examples show that the present element is free of shear locking and is insensitive to the mesh distortion, especially for the case of very thin plates. The present element can be applied to analyze plates with arbitrary geometries; it leads to reducing the computation cost. Several parameter studies are performed to show the roles of some parameters such as the power-law index, side-to-thickness ratio, boundary conditions (BCs), and variation of the plate thickness on the static bending behavior of the FGM plates.

Formulation and experimental validation of space-fractional Timoshenko beam model with functionally graded materials effects

In this study, the static bending behaviour of a size-dependent thick beam is considered including FGM (Functionally Graded Materials) effects. The presented theory is a further development and extension of the space-fractional (non-local) Euler–Bernoulli beam model (s-FEBB) to space-fractional Timoshenko beam (s-FTB) one by proper taking into account shear deformation. Furthermore, a detailed parametric study on the influence of length scale and order of fractional continua for different boundary conditions demonstrates, how the non-locality affects the static bending response of the s-FTB model. The differences in results between s-FTB and s-FEBB models are shown as well to indicate when shear deformations need to be considered. Finally, material parameter identification and validation based on the bending of SU-8 polymer microbeams confirm the effectiveness of the presented model.

On bending, buckling and free vibration analysis of 2D-FG tapered Timoshenko nanobeams based on modified couple stress and surface energy theories

In this study, the static bending, buckling, and free vibration behaviors of two-dimensional functionally graded (2D-FG) tapered micro/nanobeams are modelled and analyzed. For the first time, the nonclassical equations of motion and corresponding boundary conditions of 2D-FG beam are simultaneously derived in the framework of the modified couple stress and Gurtin-Murdoch surface elasticity theories in conjunction with the Timoshenko beam theory. Here, all the material properties of the beam are graded along the thickness and length directions according to the power-law. Both thickness and width of the beam are considered to vary linearly along the length direction. Unlike existing Timoshenko beam models, the current formulation accounts for the axial and bending deformations, Poisson’s effect, and exact position of the physical neutral plane. Differential quadrature method is employed to estimate the variable coefficients of the derived equations. The bending deflection, critical buckling load, and free vibration frequency of simply supported 2D-FG Timoshenko nanobeam are obtained employing the Navier's method. A parametric study is accompanied to investigate the effects of the different material and geometrical parameters on the mechanics of 2D-FG micro/nanobeams. It is observed that these parameters are very important in investigating the static and dynamic responses of 2D-FG micro/nanobeam.

Closed-form solution in bi-Helmholtz kernel based two-phase nonlocal integral models for functionally graded Timoshenko beams

Static bending behavior of functionally graded (FG) Timoshenko beams is studied via both strain- and stress-driven two-phase local/nonlocal mixed integral models based on the bi-Helmholtz kernel. The differential governing equations and boundary conditions are derived via the principle of minimum potential energy, the relations between strain and nonlocal stress are expressed as integral equations with the bi-Helmholtz kernel. The Laplace transform technique is utilized to obtain closed-form solutions for the first time. The numerical results are validated by reverting to the local case, and it is concluded that the present model shows well-posedness for exploring bounded beams in contrast to Eringen's pure nonlocal model: for all boundary and loading conditions, the increasing nonlocal length-scale parameter shows consistently softening or stiffening effect on the beam behaviour, according to the strain- or the stress-driven model, respectively. Moreover, the effects of shear-deformation and FG characteristics are also investigated. As the FG parameter increases, the non-dimensional deflection increases, while the softening effect caused by the strain-driven nonlocal phase seems to be more significant.

Bending and stress responses of the hybrid axisymmetric system via state-space method and 3D-elasticity theory

This research presents bending responses of hybrid laminated nanocomposite reinforced axisymmetric circular/annular plates (HLNRACP/ HLNRAAP) within the framework of non-polynomial under mechanical loading and various type of initially stresses via the three-dimensional elasticity theory. The current structure is on the Pasternak type of elastic foundation and torsional interaction. The state-space approach and differential quadrature method (SS-DQM) are studied to present the bending characteristics of the current structure by considering various boundary conditions. To predict the material properties of the bulk, the role of mixture and Halpin–Tsai equations are studied. For modeling the circular plate, a singular point is studied. Finally, a parametric study investigates the impacts of various types of distribution of laminated layers, stacking sequence on the stress/strain information of the HLNRACP/ HLNRAAP. Results reveal that the system's static stability and bending behavior improve due to increasing the value of Winkler and Pasternak factors, and the stress distribution becomes more uniform.

Finite Element Modeling for Static Bending Behaviors of Rotating FGM Porous Beams with Geometrical Imperfections Resting on Elastic Foundation and Subjected to Axial Compression

The static bending analysis of the FG porous beam resting on the two‐parameter elastic foundation is initially carried out using a combination of Reddy’s high‐order shear deformation theory and the finite element technique, where the initial geometrical imperfection and rotation movement in one fixed axis are calculated. Through the power‐law distribution function with porosities, material characteristics vary constantly from one surface to the next in the direction of thickness, and the beam is concurrently impacted by an acting force perpendicular to the beam axis and an axial compressive force. The stiffness matrix of the beam element changes as a result, and the static bending response of this beam is significantly different from that of ordinary beams. Comparison cases with published findings are used to verify the computational theory. The calculations clearly reveal many innovations for rotating beams that are influenced by many different kinds of loads, which may be used to the designing, manufacturing, and usage of these structures in reality.

Prediction of Mechanical Performance of Acetylated MDF at Different Humid Conditions

Change of relative humidity (RH) in surrounding environment can greatly affect the physical and mechanical properties of wood-based panels. Commercially produced acetylated medium density fiberboard (MDF), Medite Tricoya®, was used in this study to predict strength and stiffness under varying humid conditions by separating samples in parallel (//) and perpendicular (⊥) to the sanding directions. Thickness swelling, static moduli of elasticity (MOEstat) and rupture (MORstat), and internal bond (IB) strength were measured at three different humid conditions, i.e., dry (35% RH), standard (65% RH) and wet (85% RH). Internal bond (IB) strength was also measured after accelerated aging test. A resonance method was used to determine dynamic modulus of elasticity (MOEdyn) at the aforementioned humid conditions. Linear regression and finite element (FE) analyses were used to predict the MDF’s static bending behavior. Results showed that dimensional stability, MOEstat, MORstat and IB strength decreased significantly with an increase in RH. No reduction of IB strength was observed after 426 h of accelerated aging test. A multiple regression model was established using MOEdyn and RH values to predict MOEstat and MORstat. In both directions (// and ⊥), highly significant relationships were observed. The predicted and the measured values of MOEstat and MORstat were satisfactorily related to each other, which indicated that the developed model can be effectively used for evaluating the strength and stiffness of Medite Tricoya® MDF samples at any humid condition. Percent errors of two different simulation techniques (standard and extended FE method) showed highly efficient way of simulating the MDF structures with low fidelity.