First-order Shear Deformable Model Research Articles

Porosity-dependent wave propagation in multi-directional functionally graded nanoplate with nonlinear temperature-dependent characteristics on Kerr-type substrate

The present study investigates the wave propagation characteristics of porous, multi-directional, functionally graded (FG) nanoplates embedded in a Kerr-elastic substrate. A first-order shear deformation model (FSDM) and the nonlocal strain gradient approach (NSGA) are utilized to address this issue. The NSGA is enhanced by considering softening and hardening material effects, leading to improved accuracy in the obtained results. In accordance with the law of mixing, the material properties of the nanoplates exhibit nonlinear temperature-dependent variations along the structural axes. Both even and uneven porosity profiles are considered in this study. The nonclassical governing equations for the proposed structure are derived using Hamilton's approach. The phase velocity and wave frequency are determined through the analytical solution of an eigenvalue problem, which depends on the wave number. The dispersion properties of waves in porous FG nanoplates are examined with respect to various factors, including porosity distributions, wave numbers, nonlocal and strain gradient effects, elastic foundation coefficients, and volume fraction index. The main findings of the study indicate that increasing the volume fraction index in the thickness direction leads to a decrease in the phase velocity and frequency of wave propagation in a porous FG nanoplate for a given wave number. Additionally, regardless of the porosity pattern, an increase in the porosity coefficient results in a decrease in wave frequency and phase velocity.

Thermo-Electro-Mechanical Analysis of Micropolar FGP Cylindrical Shell Covered with Piezoelectric Actuator Layers

An investigation is performed in this paper to analyze the nonlinear thermo-electro-mechanical response of long sandwich cylindrical shells with functionally graded porous (FGP) core and thin piezoelectric actuator layers. The FGP core of the sandwich shell is assumed to be temperature- and microstructure-dependent. It is assumed that the sandwich shell is subjected to uniform lateral pressure loading in a thermo-electrical environment. The equilibrium equations of the shell with infinite length are established with the aid of the virtual displacement principle and von Kármán kinematic assumptions. The governing equations are obtained in terms of displacement components based on the first-order shear deformation model of shallow cylindrical shells and the modified couple stress theory. These nonlinear differential equations are analytically solved for a sandwich shell having both the simply-supported and clamped–clamped edge conditions by employing a two-step perturbation technique. Analytical closed-form solutions are determined as a relationship including the load parameter and the mid-span deflection. The comparison examples are made with the existing results in the literature for a simple functionally graded shell, where good agreement is obtained. It is shown that the nonlinear response of the sandwich shell is highly dependent upon the temperature variation, power law index, porosity coefficient, couple stress parameter, piezoelectric layers, and geometrical parameters of the shell.

Equivalent Single Layer Models in Free Vibration Analysis of Laminated Multi-Layered Plates

The performance of selected equivalent single-layer (ESL) models is evaluated within several classical benchmark tests for small amplitude free vibration analysis of multi-layered plates. The authors elaborated their own Finite Element software based on the first-order shear deformation (FOSD) theory with some modifications incorporated including a correction of the transverse shear stiffness and an application of zigzag type functions. Seven different ESL models were considered in the study; beside the classical FOSD model, there were three FOSD models with various transverse shear corrections and three ESL models enhanced by the application of zigzag functions and based on Reissner’s Mixed Variational Theorem.

A nonlocal strain gradient theory for rotating thermo-mechanical characteristics on magnetically actuated viscoelastic functionally graded nanoshell

In this article, a first-order shear deformable model is expanded based on the nonlocal strain gradient theory to vibration analysis of smart nanostructures under different boundary conditions. The governing equations of motion of rotating magneto-viscoelastic functionally graded cylindrical nanoshell in the magnetic field and corresponding boundary conditions are obtained using Hamilton’s principle. To discretize the equations of motion, the generalized differential quadrature method is applied. The aim of this work is to investigate the effects of the temperature changes, nonlocal parameter, material length scale, viscoelastic coefficient, various boundary conditions, and the rotational speed of this smart structure on natural frequencies of rotating cylindrical nanoshell made of magneto-viscoelastic functionally graded material.

Buckling analysis of orthotropic protein microtubules under axial and radial compression based on couple stress theory

Protein microtubules (MTs) are one of the important intercellular components and have a vital role in the stability and strength of the cells. Due to applied external loads, protein microtubules may be involved buckling phenomenon. Due to impact of protein microtubules in cell reactions, it is important to determine their critical buckling load. Considering nature of protein microtubules, various parameters are effective on microtubules buckling. The small size of microtubules and also lack of uniformity of MTs properties in different directions caused the necessity of accuracy in the analysis of these bio-structure. In fact, microtubules must be considered as a size dependent cylinder, which behave as an orthotropic material. Hence, in the present work using first-order shear deformation model (FSDT), the buckling equations of anisotropic MTs are derived based on new modified couple stress theory (NMCST). After solving the stability equations, the influences of various parameters are measured on the MTs critical buckling load.

An efficient and simple shear deformation theory for free vibration of functionally graded rectangular plates on Winkler-Pasternak elastic foundations

This work presents a simple hyperbolic shear deformation theory for analysis of functionally graded plates resting on elastic foundation. The proposed model contains fewer number of unknowns and equations of motion than the first-order shear deformation model, but the transverse shear stresses account for a hyperbolic variation and respect the tangential stress-free boundary conditions on the plate boundary surface without introducing shear correction factors. Equations of motion are obtained from Hamilton\'s principle. The Navier-type analytical solutions for simply-supported plates are compared with the existing solutions to demonstrate the accuracy of the proposed theory.

Vibration and buckling of first-order shear deformable circular cylindrical micro-/nano-shells based on Mindlin's strain gradient elasticity theory

Abstract In this research, a size-dependent first-order shear deformable model is developed based on the Mindlin's strain gradient elasticity theory to analyze the free vibration and axial buckling of circular cylindrical micro-/nano-shells. The size-dependent governing equations and corresponding boundary conditions are established through Hamilton's principle. For some specific values of the gradient-based material parameters, the most general form of shell formulation can be reduced to those based on simple forms of the strain gradient elasticity theory such as the modified strain gradient theory (MSGT), modified couple stress theory (MCST) and strain gradient theory (SGT). To illustrate the characteristics of a micro-/nano-shell obtained by the size-dependent shell formulation, the axial buckling and free vibration problems of a simply-supported (SS) microshell are analyzed by employing a Navier-type solution. Selected numerical results are presented to get an insight into the effects of dimensionless material length scale parameters, length-to-radius ratio and circumferential mode number on the non-dimensionless natural frequencies and buckling loads. For comparison purpose, the non-dimensional natural frequencies and buckling loads predicted by MSGT, MCST, SGT and classical theory (CT) are also presented. It is shown that the effect of small scale is more prominent for lower values of dimensionless length scale parameter.

Analysis of Temperature Field in a Composite Functionally Graded Material Plate by Finite Element Method

A finite element model is constructed to analyze the effects of steady state temperature field on FGM layer thickness. The first-order shear deformation model is exploited to investigate the uncoupled thermal behavior of functionally graded plates in Abacus environment. The continuum is divided into 540 elements and 541 nodes using two node linear elements. The results show that the temperature distribution in the composite plate is more reasonable with increase in the thickness of FGM layer. The comparison with the non-graded two layered composite plate, the temperature field of the Zirconia/FGM/Aluminum three layered composite plate is in the form of a curve but in case of non-graded two layered composite plate the temperature field is in the form of inclined straight line with sharp bend at the interface of metal and ceramic phase.

Zeroth-Order Shear Deformation Micro-Mechanical Model for Periodic Heterogeneous Beam-like Structures

This paper discusses a new model for investigating the micro-mechanical behavior of beam-like structures composed of various elastic moduli and complex geometries varying through the cross-sectional directions and also periodically-repeated along the axial directions. The original three-dimensional problem is first formulated in an unified and compact intrinsic form using the concept of decomposition of the rotation tensor. Taking advantage of two smallness of the cross-sectional dimension-to-length parameter and the micro-to-macro heterogeneity and performing homogenization along dimensional reduction simultaneously, the variational asymptotic method is used to rigorously construct an effective zeroth-order beam model, which is similar a generalized Timoshenko one (the first-order shear deformation model) capable of capturing the transverse shear deformations, but still carries out the zeroth-order approximation which can maximize simplicity and promote efficiency. Two examples available in literature are used to demonstrate the consistence and efficiency of this new model, especially for the structures, in which the effects of transverse shear deformations are significant.

Zeroth-order shear deformation micro-mechanical model for composite plates with in-plane heterogeneity

This article introduces a new model for investigating the mechanical behavior of heterogeneous plates, which are composed of periodically-repeated microstructures along the in-plane directions. We first formulate the original three-dimensional problem in an intrinsic form for implementation into a single unified formulation and application to geometrically nonlinear problem. Taking advantage of smallness of the plate thickness-to-length parameter and heterogeneity and performing homogenization along dimensional reduction simultaneously, the variational asymptotic method is used to rigorously construct an effective zeroth-order plate model, which is similar to a generalized Reissner–Mindlin model (the first-order shear deformation model) capable of capturing the transverse shear deformations, but still carries out the zeroth-order approximation which can maximize simplicity and promote efficiency. This present approach is incorporated into a commercial analysis package for the purpose of dealing with realistic and complex geometries and constituent materials at the microscopic level. A few examples available in literature are used to demonstrate the consistence and efficiency of this new model, especially for the structures, in which the effects of transverse shear deformations are significant.

Modeling of Progressive Damage in the Adhesive Bond Layers of Actuated Plates

This article discusses finite element modeling of progressive damage in the adhesive bond layers of actuated plates and investigates the reduction in actuation capacity caused by the damaged bond layers. The primary challenge posed by this class of problems stems from the vast range of geometric scales that are represented, with the thickness of the adhesive layer representing the smallest scale, the overall thickness of the actuated plate representing the intermediate scale, and the in-plane dimensions of the plate representing the largest scale. In multiscale problems, the overall efficiency of the numerical methodology is of paramount importance, thus model development is guided by the need to obtain a sufficiently accurate solution at an acceptable computational expense. In this study, this goal is achieved through the use of a hierarchical, displacement-based, 2-D finite element model that includes the first-order shear deformation (FSD) model, Type I layerwise models (LW1) and Type II layerwise models (LW2) as special cases. Both the LW1 layerwise model and the more familiar FSD model use a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the LW1 model includes discrete layer transverse shear effects via in-plane displacement components that are C0 continuous with respect to the thickness coordinate. The LW2 layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects by expanding all three displacement components as C0 continuous functions of the thickness coordinate. The hierarchical finite element model incorporates a 3-D continuum damage mechanics model that predicts local orthotropic damage evolution and local stiffness reduction at the geometric scale represented by the individual material ply or, in the case of layerwise models, by the individual numerical layer. The results clearly demonstrate that the resulting model can efficiently simulate progressive damage in the adhesive layers. For rectangular actuator patches, the adhesive damage is highest near the corners of the actuator and is driven primarily by local concentrations in the transverse normal and transverse shear stresses. In contrast to previous studies that have shown that the inclusion of discrete layer transverse normal stress does not significantly influence the predicted global deformations, the present study shows that the transverse normal stress has a very significant effect in the initiation and progression of localized damage in the adhesive layers.

The Effect of Laminate Kinematic Assumptions on the Global Response of Actuated Plates

The effects of accurately accounting for transverse shear strain, transverse normal strain, and discrete layer kinematics on the computed global response of actuated plates are investigated using a hierarchical displacement-based, two-dimensional (2-D) finite element model that is developed specifically for composite laminates. The hierarchical model is used to obtain the first-order shear deformation model, a higher-order cubic equivalent-singlelayer model, a type-I layerwise model, and a type-II layerwise model as special cases. Each of the first three models uses a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the models differ significantly in their assumed distribution of transverse shear strain. The type-II layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects. The scope of the study is restricted to homogeneous plates with surface-mounted actuators covering a wide range of span-to-thickness ratios. The results clearly demonstrate that as the span-to-thickness ratio of the actuated region decreases, discrete layer kinematics become very important for accurately characterizing the global response of the actuated plate. The set of laminate kinematic assumptions utilized by a particular model influence the predicted global response primarily through the so-called local kinematic effect where a portion of the available actuation energy is diverted to the production of localized transverse shear deformation and localized transverse normal deformation in the vicinity of the actuator edges, thereby reducing the amount of actuation energy that is available for producing the intended global deformation mode.

Layerwise modeling of progressive damage in fiber-reinforced composite laminates

This paper investigates the effects of discrete layer transverse shear strain and discrete layer transverse normal strain on the predicted progressive damage response and global failure of fiber-reinforced composite laminates. These effects are isolated using a hierarchical, displacement-based 2-D finite element model that includes the first-order shear deformation model (FSD), type-I layerwise models (LW1) and type-II layerwise models (LW2) as special cases. Both the LW1 layerwise model and the more familiar FSD model use a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the LW1 model includes discrete layer transverse shear effects via in-plane displacement components that are C 0 continuous with respect to the thickness coordinate. The LW2 layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects by expanding all three displacement components as C 0 continuous functions of the thickness coordinate. The hierarchical finite element model incorporates a 3-D continuum damage mechanics (CDM) model that predicts local orthotropic damage evolution and local stiffness reduction at the geometric scale represented by the homogenized composite material ply. In modeling laminates that exhibit either widespread or localized transverse shear deformation, the results obtained in this study clearly show that the inclusion of discrete layer kinematics significantly increases the rate of local damage accumulation and significantly reduces the predicted global failure load compared to solutions obtained from first-order shear deformable models. The source of this effect can be traced to the improved resolution of local interlaminar shear stress concentrations, which results in faster local damage evolution and earlier cascading of localized failures into widespread global failure.

Accurate determination of transverse normal stresses in sandwich panels subjected to thermomechanical loadings

A two-stage computational procedure is presented for the accurate determination of transverse normal stresses in sandwich panels subjected to thermomechanical loadings. The procedure involves the use of a first-order shear deformation model in the first stage, and an iterational process for successive improvement of the accuracy of the displacement and stress fields in the second stage. The effectiveness of the procedure is demonstrated by means of numerical studies of thin and moderately thick flat rectangular sandwich panels. Two sets of boundary conditions are considered; namely, one with all edges simply supported, and the other with two opposite edges simply supported and the remaining two clamped. The displacement components and the transverse shear and normal stresses obtained by the proposed computational procedure are found to be in close agreement with the solutions of the three-dimensional (3-D) continuum models.