Higher-order Shear Deformation Theory Research Articles

The Nonlinear Bending of Sector Nanoplate via Higher-Order Shear Deformation Theory and Nonlocal Strain Gradient Theory

In this context, the nonlinear bending investigation of a sector nanoplate on the elastic foundation is carried out with the aid of the nonlocal strain gradient theory. The governing relations of the graphene plate are derived based on the higher-order shear deformation theory (HSDT) and considering von Karman nonlinear strains. Contrary to the first shear deformation theory (FSDT), HSDT offers an acceptable distribution for shear stress along the thickness and removes the defects of FSDT by presenting acceptable precision without a shear correction parameter. Since the governing equations are two-dimensional and partial differential, the extended Kantorovich method (EKM) and differential quadrature (DQM) have been used to solve the equations. Furthermore, the numeric outcomes were compared with a reference, which shows good harmony between them. Eventually, the effects of small-scale parameters, load, boundary conditions, geometric dimensions, and elastic foundations are studied on maximum nondimensional deflection. It can be concluded that small-scale parameters influence the deflection of the sector nanoplate significantly.

PHÂN TÍCH TĨNH VỎ TRỤ FG-CNTRC CHỊU TẢI TRỌNG CƠ NHIỆT VỚI TÍNH CHẤT VẬT LIỆU PHỤ THUỘC VÀO NHIỆT ĐỘ

This article presents a thermo-elastic static analysis of the FG-CNTRC cylinderical shell subjected to a thermo-mechanical load. The temperature is assumed to vary through the thickness, and the temperature distribution is derived from the heat transfer formula. The gorvening equations of thermoelastic analysis are established by using the higher-order shear deformation theory, including the transverse normal stress effect. The material properties of FG-CNTRC are temperature-dependent. Using an analytical solution that used the single trigonometric series and the Laplace transformation to solve these equations of shell with the different boundary conditions. The present approach is validated by comparing the results with published results. The effects of different boundary conditions and temperatures on the displacement and stresses of the FG-CNTRC cylindrical shells are investigated. The results show the jumping variation of the stress components at the boundary position. Besides, when the temperature increases, the maximum absolute value of the axial stress increases while that of the circumferential stress decreases.

Linear finite element formulation for free vibration and buckling analyses of multi-directional FGP doubly curved shallow shells in thermal environment

ABSTRACT This paper presents the free vibration and buckling analyses of multi-directional functionally graded porous doubly curved shallow shells resting on Pasternak elastic foundations in a thermal environment. For the first time, a linear finite element formulation using the refined high-order shear deformation theory, combined with the Hermitian function, is developed to compute the vibration and buckling behaviour of doubly curved shallow shells. Mechanical properties of MFGP material change according to the length, width and thickness directions and the porosity distribution including even and uneven. The doubly curved shallow shell is operated in uniform, linear and non-linear temperature environments. The convergence and accuracy of the proposed method are verified by comparing numerical results with published works. In addition, a comprehensive numerical investigation was carried out to evaluate the influence of parameters on the natural frequency and critical buckling behaviour of MFGP doubly curved shallow shells.

NGHIÊN CỨU PHÂN BỐ ỨNG SUẤT TẠI BIÊN NGÀM CỦA VỎ TRỤ FGM CHỊU TẢI TRỌNG TĨNH SỬ DỤNG LÝ THUYẾT BIẾN DẠNG CẮT BẬC CAO QUASI-3D

This article presents the results of investigating FGM closed cylindrical shells using quasi-3D higher-order shear deformation theory. The law of volume distribution of materials according to the power function. In this article, the results of the analysis of the stress state at the clamped boundary area of the FGM cylindrical shell subjected to radial local loads distributed according to different laws. The mathematical model and the calculation program in Maple are compared and verified with the published results. The article has shown that at the clamped boundary edge, the values of stress components increase sharply, and at the same time, the strong influence of the load parameters (the variation law, the size of the load area), the power-law index on the values and distribution of stress components in the boundary area is demonstrated.

Reliability and sensitivity analyses of porous functionally graded graphene platelet reinforced composite plate

The performance of composite is seriously influenced by numerous uncertain factors, in which the uncertainty analysis plays a vital role in composite materials analysis by providing a comprehensive understanding of uncertain factors and their influence on design and performance. In this study, uncertain behaviors of porous functionally graded graphene platelets reinforced composite (FG-GPLRC) plates are first investigated for assessing the safety under bending deformation. The mechanical model for static analysis of the porous FG-GPLRC plate is formulated by Reddy’s higher-order shear deformation theory, and the bending responses are derived by Hamilton’s principle and the Navier method. Material properties and loads are deemed as random variables, and the performance function is determined by the Tsai-Wu criterion. The univariate dimension reduction method and maximum entropy method are employed to estimate the reliability of porous FG-GPLRC plates with diverse distribution patterns. Additionally, the analytical sensitivities of failure probability with respect to material parameters are derived. The analysis results illustrate that the existence of uncertain factors significantly influences the performance and reliability of composites, and the structural reliability can be improved by adding appropriate pores and graphene platelets at the edge layer of the composite plate.

Nonlinear forced vibration of functionally graded hybrid three-phase nanocomposite toroidal shell segments reinforced by carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs)

This study proposes a unique three-phase functionally graded (FG) hybrid nanocomposite material reinforcing the toroidal shell segments to investigate the nonlinear forced vibration using Reddy's higher-order shear deformation theory, von Karman geometrical nonlinearity and Stein-McElman's assumption along with the Hamilton principle. The examined toroidal shell's material consists of polymeric resin, carbon nanotubes (CNTs), and graphene platelets (GPLs) as nanocomposites. The material properties have been derived based on a modified Halpin-Tsai micromechanical model. Four distribution patterns have been studied: uniform (UD), FG-X, FG-O, and FG-V. The continuous distribution of GPLs and CNTs leads to inhomogeneous position-dependent properties throughout the shell thickness. The obtained differential equations of motion have been reduced to ordinary equations using Galerkin's technique. A multi-scales method (MSM) is used to estimate a closed-form solution representing the frequency-amplitude relation, and the state space representation is used along with numerical fourth-order Runge Kutta method (RK4) to obtain the nonlinear dynamic response of the toroidal shell. The accuracy of the current results obtained has been verified by comparing them with the relevant literature and numerical results using (RK4). In addition, the influence of both GPLs and CNTs weight fractions, nanofillers’ distribution types, longitudinal and circumferential wave numbers, elastic foundation parameters, static axial compression load, transverse excitation load, damping ratio, geometrical characteristics of the toroidal shell on the dimensionless natural frequency, nonlinear primary resonance (frequency-amplitude curve), and nonlinear dynamic response are carefully studied. The results found that combining (GPLs & CNTs) into the shell's matrix improves performance, especially in concave shells compared to convex ones. The FG-X pattern reduces peak resonance amplitude and improves natural frequency. A reduction of 12.3% is observed for convex shells, and 21.5% for concave shells compared to FG-O.

Linear static, geometric nonlinear static and buckling analyses of sandwich composite beams based on higher-order refined zigzag theory

This study aims at developing finite element formulations based on higher-order refined zigzag theory (HRZT) to analyze sandwich composite beams under linear static bending, geometric nonlinearity and buckling. In HRZT, the higher-order terms associated with average shear strain and rotation due to zigzag are added based on refined zigzag theory (RZT). In linear static bending analysis, both C0 and C1 elements are developed. For the C0 element, an extra node is used in order to eliminate shear locking. For geometric nonlinear static analysis, the two-node C0 element is developed to solve the bending response with lateral loading and buckling response. The numerical results are compared with those calculated by 2D exact solutions, analytical solutions of HRZT and 2D models using commercial software. Various beam theories such as first-order shear deformation theory (FSDT), higher-order shear deformation theory (HSDT), RZT, and other types of higher-order zigzag theory are also introduced for comparisons. Through comprehensive comparisons with existing works on FEM or zigzag theories, the HRZT beam elements developed in this study exhibit superior accuracy and are free from shear locking in linear static analysis. Furthermore, the study also demonstrates that these elements achieve high accuracy in geometric nonlinear analysis and buckling analysis.

A new set of explicit solutions for postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution and thermal loading

Thermal postbuckling analysis is presented for shear-deformable anisotropic laminated cylindrical shell with general imperfection distribution under several typical temperature fields are analyzed, including uniform temperature, tent-like linear distribution, axial and circumferential quadratic parabolic temperature fields over the shell surface and through the shell thickness. On this basis, the general temperature distribution can be obtained based on the three-dimensional steady-state heat transfer equation. At the same time, based on the high order shear deformation theory including anisotropic coupling effect, geometrically nonlinear large deflection displacement-strain relationship, the general initial geometric imperfections obtained by measurement and analytical means are introduced, and the thermal buckling equilibrium differential equation of anisotropic shear deformable laminated cylindrical shell with initial imperfections is established. A two-step perturbation-Galerkin integral method is used to obtain explicit solutions for the path development direction of post-thermal buckling equilibrium. The results of parameter analysis show that the material properties and temperature dependence, temperature distribution, lay-up and geometric parameters such as radius-to-thickness ratio have important effects on the thermal buckling load and postbuckling equilibrium path. The present method provides an effective theoretical prediction and analysis method for the design of the carrying capacity of submarine pipelines.

Analysis of doubly curved laminated composite shells using hybrid-Trefftz finite element model based on a high order shear deformation theory

Static analysis of doubly curved laminated composite shells with rectangular plan form has been carried out using a novel hybrid Trefftz finite element (HTFE) model. A typical higher order shear deformation theory (HSDT) has been used for describing the kinematics of deformations delineating both internal and auxiliary displacement fields. This HSDT allows to derive the system of governing partial differential equations of the single layer equivalent to the element domain. The characteristic functions forming the exact solutions of these homogenous governing equilibrium equations are used as the Trefftz functions. A key aspect of our HTFE strategy is its efficiency in bypassing the use of particular solution of the governing equations. Validation against the exact solutions reveals the precision of this novel HTFE model. The versatility of the Trefftz functions allows them to accommodate a spectrum of geometric configurations, ranging from standard to complex polygonal HTFE designs. The HTFE model for the doubly curved composite shells employing the HSDT marks a significant advancement in the analysis of doubly curved spherical, paraboloid and hyperboloid cross-ply and antisymmetric angle-ply shells.

Nonlinear dynamic buckling analysis of the FG-GPLRC cylindrical and sinusoid panels with porous core under time-dependent axial compression

The nonlinear dynamic buckling responses of functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and sinusoid panels with porous core are presented in this paper. The governing formulations are established by applying the nonlinear higher-order shear deformation theory (HSDT). an approximation technique is used to determine the stress function for complexly curved panels. Euler-Lagrange equations can be used to achieve the nonlinear motion equation. Numerical investigations are considered using the Runge-Kutta method for dynamic responses, and using the Budiansky-Roth criterion for critical dynamic buckling loads. Some discussions on the dynamic buckling responses of panels with porous core can be achieved from the numerical examples.

Nonlinear Vibration of Saturated Porous Functionally Graded Plate in Thermal Environment

In this work we studied investigates the nonlinear vibrations of a rectangular saturated porous functionally graded (FG) plate on elastic foundations in thermal environment. The mechanical properties of the saturated porous material vary smoothly with the thickness in three different distributions of porosity including uniform, symmetrically irregular, and asymmetrically irregular. The basic equations are employed by the Reddy’s higher order shear deformation theory, incorporating the geometrically nonlinear von Kármán strain-displacement relationship, stress-strain relations based on the elastic theory for porous materials by Biot, and an analytical solution obtained through the Galerkin method and Airy’s stress function for the simply supported plate. The influence of the geometrical and material parameters, elastic foundations and temperature increment on the nonlinear vibrations of the saturated porous FG plate were specifically evaluated through numerical investigations. 
  
  
  

Simplified homogenization technique for nonlinear finite element analysis of in-plane loaded masonry walls

The present investigation focuses on the nonlinear modelling of masonry walls under in-plane loading. To conduct this macro-scale study, a homogenization technique is applied to derive the equivalent secant moduli according to damage and friction sliding at the micro scale level for both bed mortar joint and brick material constituents. In the homogenization process the representative unit cell of masonry panel with a regular arrangement is first defined. Then a simplified kinematical model is proposed to describe the behaviour of the homogenized unit-cell under combined axial and shear stresses. This is based on the cohesive model for pure mode I and II and by allowing for the Coulomb's friction law for bed joint and the scalar damage model is adopted for the brick material. In the first stage, the proposed homogenization technique is verified for various unit cell constituents at micro-scale level. In the second step, this homogenization procedure was incorporated into proposed in-house finite elements software. To achieve this end, a new beam finite element is formulated in the context of the fibre higher order shear deformation theory HSDT. Using the proposed finite elements model, a numerical investigation is conducted to check the validity of the proposed model against the available experimental and numerical investigations. This allowed for the study of the effect of the applied compressive load on the lateral ultimate load.

Nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator: Introducing machine learning approach to measure nonlinear phase velocities

In today's rapidly changing corporate environment, business executives are turning to cutting-edge technology to solve complex problems efficiently. A machine learning approach allows for a more thorough understanding and quantification of decision-making outcomes, leading to increased confidence in business decisions. In this work for the first time, machine learning approach and mathematical simulation are used to model nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator. Using mathematical modeling, the current work studies the nonlinear phase velocity in a sandwich nanoshell composed of a bi-directional functionally graded (BD-FG) core and piezoelectric patch. The modeling uses first-order shear deformation theory with the fairly thick hypothesis to describe the displacement fields of piezoelectric layers. For core layer, a higher order shear deformation theory with terms of higher order in the Taylor series expansion is presented. The nanosystem is accurately modeled using nonlocal strain gradient theory, incorporating nonlocal and length scale characteristics. The solution approach section outlines the multiple scale method for the time domain and the harmonic solution methodology for the displacement domain to address nonlinear equations. To reduce the computational expenses, a machine-learning approach is introduced to address the nonlinear vibration issue in the current study. Validating the conclusions of the current study by comparing them with machine-learning solutions and another published paper can confirm their accuracy. The results section presents the impacts of factors like area of piezoelectric patch, length scale, applied voltage, nonlocality, geometry condition and FG power index on the nonlinear phase velocity of the BD-FG nanoshell coupled with piezoelectric patch.

Multifield and higher-order analysis of sandwich smart curved beams made of graphene origami auxetic metamaterial

This article is organized to investigate effect of material and geometric characteristics of graphene origami as well as multifield loading on the electro-magneto-elastic bending results of the sandwich smart nanocomposite curved beam. The analysis is performed using the higher-order shear deformation theory accounting out of plane normal strain. The constitutive relations are extended in the cylindrical coordinate in the presence of thermal, electrical, and magnetic loads. The governing equations are derived using the principle of virtual work with accounting electric and magnetic works. The effective material properties of graphene origami reinforced composite core are evaluated through Halpin–Tsai micromechanical models and rule of mixture for modulus of elasticity, Poisson’s ratio, and density. The analytical solution is presented for simply supported boundary conditions based on trigonometric functions. A verification test is performed for validation of the formulation, solution procedure, and numerical results. The stress and deformation components are evaluated with changes of multifield loading and characteristics of graphene origami along the thickness direction. The results show a significant effect of electrical, thermal, and magnetic loading on the stress and deformation results.

In-Plane Vibrations of Deep Sandwich Arches with Different End Conditions Based on a Logarithmic Shear Deformation Theory

This paper deals with the vibrational frequencies of deep sandwich arches to enhance their application domain and possibly use them for energy harvesting. The circular arch with porous nanocomposite core and titanium alloy face sheets having different end conditions is numerically analyzed. The middle core of the sandwich arch is made of a six-layered porous aluminum reinforced with graphene nanoplatelets. The kinematic equations are formulated in this study based on the higher-order shear deformation theory using a logarithmic function of radius. The effective properties of the nanocomposite media are modeled by employing the Halpin–Tsai modified rule. The equations of motion are determined by applying the principle of virtual displacement. The partial differential equations are reduced using the generalized differential quadrature technique to solving an algebraic eigenvalue problem. Novel numerical results are given to show the effects of geometrical parameters, material properties, and boundary conditions on the vibrations of deep sandwich arch.

A nonlocal higher-order shear deformation approach for nonline ar static analysis of magneto-electro-elastic sandwich Micro/Nano-plates with FG-CNT core in hygrothermal environment

In this study, the nonlocal static analysis of Magneto-Electro-Elastic (MEE) sandwich micro/nano-plates with Functionally Graded Carbon Nanotubes (FG-CNTs) core in a hygrothermal environment is studied. The nonlocal static formulation is developed based on Reddy's High-order Shear Deformation theory (HSDT). The von Kármán nonlinear strains are used and the governing equations of the HSDT are derived accounting for Eringen's nonlocal stress-gradient model. The governing equations are solved using the Galerkin method, which involves expanding the displacement field based on orthogonal functions. The variation of the volume fractions through the thickness of FG-CNTs core layer has been computed using two different homogenization techniques: the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the characteristics of the FG-CNTs core, nonlocal parameters, thermal and moisture changes, types of CNT reinforcement and volume fraction, electric and magnetic potentials, and geometric parameters. The results are compared with the first-order shear deformation theory (FSDT) and classical plate theory to show the accuracy of nonlocal High-order Shear Deformation theory(NHSDT) formulation.

Bulk wave propagation analysis of imperfect FG bio-composite beams resting on variable elastic medium

Bio-composites are outstanding alternatives to prevalent biomaterials in medical instruments due to their supreme material properties. On the other hand, mechanical analyses of bio-systems should be carried out to optimize the designed biomedical applications such as medical implants. Therefore, bulk wave propagation analysis of functionally graded (FG) bio-composite beams could serve as design targets of biomedical structures. Some applications like medical implants have a layered structure and FG composites will be an appropriate choice for constitutive material of such structures. In the current study, the influence of imperfection on the propagation of bulk waves in FG bio-composite beams with biomedical applications lying on a variable elastic medium has been evaluated for the first time. The constitutive materials of FG bio-composite beams are hydroxyapatite and gold alloy which are common materials in the biomedical industry. To determine the effective material properties of the structure under study, the conventional porosity-dependent homogenization scheme has been implemented. Various distribution patterns have been accounted for variable elastic medium. Moreover, a refined higher-order shear deformation theory and Hamilton’s principle have been exerted to obtain the partial differential governing equations and the derived equations are analytically solved via a harmonic function. To check the accuracy of the used methodology, the obtained result has been validated firstly with former investigations. Eventually, the sensitivity of various considerable parameters has comprehensively been probed and discussed. It is found that some parameters including wave number, power law index, and elastic foundation have positive effects and some parameters including slenderness ratio, ξ parameter, and imperfection coefficient have negative effects on the propagated bulk waves. Also, the sinusoidal pattern possesses less increasing effect than other ones.

Behavior of functionally graded carbon nanotube reinforced composite sandwich beams with pultruded GFRP core under bending effect

A novel generation of composite sandwich beams with laminated carbon fiber-reinforced polymer skins and pultruded glass fiber-reinforced polymer core materials was examined for their flexural behavior. The strength and failure mechanisms of the composite sandwich beams in flatwise and edgewise configurations were investigated using three-point static bending tests. These sophisticated composite structures must be designed and used in a variety of sectors, and our research provides vital insights into their performance and failure patterns. In comparison to the reference specimens (FGM-1), the carbon nanotube-reinforced specimens’ bending capacity was affected and ranged from −2.5% to 7.75%. The amount of the carbon nanotube addition had a substantial impact on the beams’ application level and load-carrying capacity. Particularly, the application of 0.5 wt% additive in the outermost fiber region of the beams, such as in FGM-4, led to an increase in the bending capacity. However, the stiffness values at the maximum load were decreased by 0.3%–18.6% compared to FGM-1, with the minimum level of the decrease in FGM-4. The experimental results were compared with the theoretical calculations based on the high-order shear deformation theory, which yielded an approximation between 11.99% and 12.98% by applying the Navier’s solution.

Nonlinear vibrations of porous-hyperelastic cylindrical shell under harmonic force using harmonic balance and pseudo-arc length continuation methods

The resonant responses are investigated for the porous-hyperelastic Mooney–Rivlin cylindrical shell subjected to a radial harmonic excitation. Considering the higher-order shear deformation theory (HSDT), fourth-order strain-displacement relations are derived, which include the radial geometric imperfections varying along the thickness direction. Using the porous-hyperelastic Mooney–Rivlin constitution relation and Lagrange equation, the differential governing equations of motion are obtained for the porous-hyperelastic cylindrical shells. The resonant conditions are presented and the accuracy of the mathematical models is verified. The harmonic balance and pseudo-arc length continuation methods are used to obtain the amplitude-frequency and forced-amplitude curves. The stability of the solutions is analyzed by Floquet theory. The effects of the external excitation amplitudes, structure parameters and porosity infill parameters on the linear frequencies, amplitude-frequency responses and force-amplitude responses are discussed for the imperfect porous-hyperelastic cylindrical shells. The results show that the linear frequencies of the porous-hyperelastic cylindrical shells increase obviously as the uniform and sigmoid function porosity parameters reach the certain values. The increase of the structure parameters enhances the response amplitudes of the first-order mode and minified response amplitudes of the second-order mode. The decreasing porosity ratios weaken the softening nonlinear behaviors of the porous-hyperelastic cylindrical shells. With the changes of the external excitation amplitudes and structure parameters, the motion of the porous-hyperelastic cylindrical shell indicates that the synchronous vibrations occur with the period and chaotic vibrations alternately.

Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate–Shell Panels

An analysis of the influence of applied loads on free vibrations of plate and shell panels made of functionally graded materials is analyzed in the present work. Formulations for the static analysis considering geometrically nonlinear behavior, as well as linear buckling and free vibrations analyses are considered. A calculation of the through-thickness stress distribution is also performed. The finite element model is based on a higher order shear deformation theory using a non-conforming flat triangular plate/shell element with three nodes, and eight degrees of freedom per node is used in the numerical implementation. The results obtained with this numerical model are presented, discussed, and compared with alternative solutions published by other authors in some benchmark applications.